Inclusion Criteria:

          -  Histologically or cytologically confirmed thyroid carcinoma of follicular origin
             (including papillary, follicular, and poorly differentiated subtypes and their
             respective variants).

          -  A tumor sample (primary, recurrent, or metastatic tumors) possessing a BRAF V600
             mutation, as confirmed in a CLIA-certified laboratory or using an FDA-approved assay

          -  Measurable disease by RECIST v1.1 (tumors in previously irradiated fields may be
             considered measurable if there is evidence of tumor progression after radiation
             treatment)

          -  RAIR disease, as defined by any one of the following:

               -  A metastatic lesion that is not RAI-avid on a diagnostic radioiodine scan

               -  An RAI-avid lesion that remained stable in size or progressed despite RAI
                  treatment before entry in this study (there are no size limitations for the index
                  lesion used to satisfy this entry criterion)

               -  The presence of at least 1 FDG-avid lesion

          -  No receipt of treatment for thyroid cancer, defined as:

               -  No I-131 therapy < 6 months before initiation of the protocol (time of initiation
                  of the protocol is defined as the first day of drug therapy with vemurafenib and
                  copanlisib); diagnostic activities of I-131 (0-10m Ci) are allowed within 6months
                  of initiating the protocol

               -  No external beam radiation therapy <4 weeks before initiation of the protocol

               -  No chemotherapy or targeted therapy including TKIs <4 weeks (or <5 half lives of
                  the drug) before the initiation of this protocol

          -  Age of ≥ 18 years

          -  ECOG performance status ≤ 2 or Karnofsky Performance Score (KPS) ≥ 70%

          -  Tissue from the primary tumor or metastases available for correlative studies. Either
             a paraffin block or at least 20 unstained slides are acceptable (30 unstained slides
             is ideal); if <20 unstained slides are available, and a paraffin block is not
             available, the patient may be able to participate at the discretion of the
             investigator

          -  Able to swallow and retain an orally administered pill without any clinically
             significant gastrointestinal abnormalities that may alter absorption, such as
             malabsorption syndrome or major resection of the stomach or bowels

          -  Agree to undergo 2 research biopsies of (a) malignant lesion(s). Tumor tissue obtained
             before study consent or treatment can also be submitted in lieu of performance of the
             first pretreatment biopsy if the Principal Investigator deems it to be of sufficient
             quantity/quality/timeliness (tumor tissue obtained more than 3 years from time of
             study consent would not be eligible). Patients may be exempt from biopsy if (1) the
             investigator or person performing the biopsy judges that no tumor is accessible for
             biopsy, (2) the investigator or person performing the biopsy feels that the biopsy
             poses too great of a risk to the patient, or (3) the patient's platelet count is
             <100,000/mcL or the patient cannot be safely removed from anticoagulation therapy (if
             the anticoagulation therapy needs to be temporarily held for the biopsy procedure). If
             the investigator deems a second research biopsy to be high risk, the patient may be
             exempt from the second biopsy.

          -  Screening laboratory values meeting the following criteria:

               -  WBC ≥ 2000/µL

               -  Neutrophils ≥ 1500/µL

               -  Platelets ≥ 100 × 10^3/µL

               -  Hemoglobin >9.0 g/dL

               -  Lipase ≤ 1.5 × ULN

               -  AST/ALT ≤ 3 × ULN

               -  Total bilirubin ≤ 1.5 x ULN (except subjects with Gilbert syndrome, who can have
                  total bilirubin <3.0 mg/dL)

               -  Serum creatinine ≤ 1.5 x ULN or creatinine clearance (CrCl) ≥ 40 mL/min (if using
                  the Cockcroft-Gault formula below):

        Female CrCl = (140 - age in years) x weight in kg x 0.85 / 82 x serum creatinine in mg/dL

        Male CrCl = (140 - age in years) x weight in kg x 1.00 / 72 x serum creatinine in mg/dL

        Exclusion Criteria:

          -  Untreated metastatic brain or leptomeningeal tumors (metastatic brain or
             leptomeningeal tumors treated with radiation and/or surgery are allowed)

          -  Prior malignancy if diagnosed and treated within 2 years of trial drug initiation
             (with the exception of nonmelanoma skin cancers or Stage I cancers treated with
             curative intent).Patients may be included if they have completed therapy for a prior
             malignancy >2 years before drug initiation and currently have no evidence of disease

          -  Inability to follow a low-iodine diet or requiring a medication with a high content of
             iodide (amiodarone)

          -  Current congestive heart failure class >2, as defined by the New York Heart
             Association functional classification system

          -  Myocardial infarction < 6 months before the initiation of protocol

          -  Unstable angina (angina symptoms at rest) or new-onset angina (begun within the last 3
             months)

          -  Uncontrolled hypertension (blood pressure >150/90, despite optimal medical management)

          -  Uncontrolled type I or II diabetes mellitus, as judged by the investigator, or Hgb A1C
             of >8.5

          -  Arterial or venous thrombotic event or embolic event, such as a cerebrovascular
             accident (including transient ischemic attacks), deep vein thrombosis, or pulmonary
             embolism, within 3 months before the start of study medication

          -  Nonhealing wound, ulcer, or bone fracture (tumor-related nonhealing wounds are
             allowed)

          -  Active, clinically serious infections CTCAE v5.0 grade >2

          -  History of concurrent condition of interstitial lung disease and/or severely impaired
             lung function

          -  Known history of HIV infection (all patients must be screened for HIV up to 28 days
             before start of study)

          -  Seizure disorder requiring medication

          -  Therapy with a prohibited concomitant medication that cannot be temporarily held (at
             least 2 weeks before initiation of vemurafenib plus copanlisib until 1 week after the
             last dose) or replaced with a nonprohibited concomitant medication

          -  Systemic corticosteroid therapy at a daily dose >15 mg prednisone or equivalent
             (previous corticosteroid therapy must be stopped or reduced to the allowed dose at
             least 7 days before study registration)

          -  Cytomegalovirus (CMV) PCR-positive at baseline

          -  Evidence or history of a bleeding diathesis or any hemorrhage or bleeding CTCAE v5.0
             grade ≥ 3 within 4 weeks before the start of study protocol

          -  HBV or HCV infection. All patients must be screened for HBV and HCV up to 28 days
             before the start of study medication using the routine hepatitis virus laboratorial
             panel. Patients positive for HBsAg or HBcAb will be eligible if they are negative for
             HBV-DNA (these patients should receive prophylactic HBV antiviral therapy). Patients
             positive for anti-HCV antibody will be eligible if they are negative for HCV-RNA

          -  Known hypersensitivity to any of the test drugs, test drug classes, or excipients in
             the formulation

          -  Substance abuse or medical, psychological, or social conditions that may interfere
             with the patient's participation in the study or evaluation of the study results

          -  Patients who are pregnant

          -  Women of child-bearing potential, defined as all women physiologically capable of
             becoming pregnant, unless they are using highly effective methods of contraception
             during dosing and for 6 months after the last administration of study treatment.
             Highly effective contraception methods include:

               -  Total abstinence (when this is in line with the preferred and usual lifestyle of
                  the patient). Periodic abstinence (e.g., calendar, ovulation, symptothermal,
                  postovulation methods) and withdrawal are not acceptable methods of contraception

               -  Female sterilization (have had surgical bilateral oophorectomy with or without
                  hysterectomy), total hysterectomy, or tubal ligation at least 6 weeks before
                  taking study treatment. In cases of oophorectomy alone, only when the
                  reproductive status of the woman has been confirmed by follow-up hormone-level
                  assessment (FSH level in the postmenopausal range) is this acceptable

               -  Male sterilization (at least 6 months before screening). The vasectomized male
                  partner should be the sole partner for that patient

               -  Use of oral, injected, or implanted hormonal methods of contraception or
                  placement of an intrauterine device or intrauterine system or other forms of
                  hormonal contraception that have comparable efficacy (failure rate <1%)-for
                  example, hormone vaginal ring or transdermal hormone contraception. Note: In
                  cases of use of oral contraception, women should have been stable, on the same
                  pill for a minimum of 3 months before taking study treatment

               -  Women are considered postmenopausal and not of child-bearing potential if they
                  have had 12 months of natural (spontaneous) amenorrhea with an appropriate
                  clinical profile (i.e., age appropriate, history of vasomotor symptoms) or have
                  had surgical bilateral oophorectomy (with or without hysterectomy), total
                  hysterectomy, or tubal ligation at least 6 weeks ago. In cases of oophorectomy
                  alone, only when the reproductive status of the woman has been confirmed by
                  follow-up hormone-level assessment is she considered not of child-bearing
                  potential.

          -  Sexually active men, unless they use a condom during intercourse while on treatment
             and for 6 months after stopping treatment with study drugs (men should not father a
             child in this period). A condom is required to be used by vasectomized men as well
             during intercourse to prevent delivery of the drug via semen.