Inclusion Criteria:

* Diagnosis of Richter syndrome (RS; transformed CLL), or indolent NHL (follicular lymphoma \[FL\], lymphoplasmacytic lymphoma \[LPL\], marginal zone lymphoma \[MZL\]) in transformation. Only patients who have diffuse large B-cell lymphoma (DLBCL) histology are eligible in transformation are eligible (for example, patients with transformation into Hodgkin lymphoma subtype are not eligible).
* Participants with RS must have received at least 2 cycles of prior systemic therapy for either RS or underlying CLL.
* Participants with FL and other indolent lymphomas in transformation must have underwent \>= 1 prior chemo-immunotherapy regimen (e.g., rituximab/cyclophosphamide/doxorubicin/prednisone/vincristine \[R-CHOP\] or similar) administered for \>= 2 cycles and have had either documented disease progression to the most recent treatment regimen, or refractory disease and must not be candidates for or planning to pursue autologous stem cell transplant, or must have relapsed following autologous stem cell transplant which took place at least 3 months prior to study therapy.
* Radiographically measurable lymphadenopathy (\>= 1.5 cm) or measurable extra-nodal disease.
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2.
* Total bilirubin =\< 2 x institutional upper limit of normal (ULN).
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than 2.5 x institutional ULN.
* Estimated creatinine clearance (CrCL) using the Cockroft-Gault equation \>= 30 mL/min.
* Platelets \>= 75,000/mm\^3 (\>= 25,000/mm\^3 if due to disease involvement in the bone marrow; transfusion is not permitted to achieve this level).
* Absolute neutrophil count \>= 1000/mm\^3 (\>= 500/mm\^3 if due to disease involvement in the bone marrow).
* Female participants who:

* Are postmenopausal for at least 1 year before the screening visit, OR
* Are surgically sterile (i.e. tubal ligation), OR
* Participants of childbearing potential must have a negative serum beta-human chorionic gonadotropin at screening and:

* Agree to practice 1 highly effective method and 1 additional effective (barrier) method of contraception at the same time, from the time of signing the informed consent through 1 month after the last dose of copanlisib, or 5 months after the last dose of nivolumab, whichever is later, or
* Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.).
* Male patients, even if surgically sterilized (i.e., status post-vasectomy) must:

* Agree to practice effective barrier contraception during the entire study treatment period and through 1 month after the last dose of copanlisib, OR
* Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods for the female partner\] withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.).
* Ability to understand and the willingness to sign a written informed consent document.

Exclusion Criteria:

* History of allogeneic bone marrow or organ transplant.
* Prior therapeutic intervention with any of the following:

* Therapeutic anti-cancer antibodies within 2 weeks.
* Radio- or toxin-immunoconjugates within 10 weeks.
* Radiation therapy within 2 weeks.
* All other chemotherapy within 3 weeks prior to initiation of study therapy.
* Targeted therapy - within 6 half-lives (for example, 36 hours for ibrutinib).
* History of prior malignancy except:

* Malignancy treated with curative intent and no known active disease present for \>= 2 years prior to initiation of therapy on current study.
* Adequately treated non-melanoma skin cancer or lentigo maligna (melanoma in situ) without evidence of disease.
* Adequately treated in situ carcinomas (e.g., cervical, esophageal, etc.) without evidence of disease.
* Asymptomatic prostate cancer managed with "watch and wait" strategy.
* Any adverse event related to prior therapy that has not recovered to =\< grade 1 (excluding grade 2 alopecia and grade 2 neuropathy).
* Chronic use of corticosteroids in doses which exceed 15 mg of prednisone per day, or the equivalent.
* Uncontrolled immune hemolysis or thrombocytopenia.
* A history of human immunodeficiency virus (HIV) infection. HIV-positive participants on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with copanlisib and/or nivolumab. In addition, these participants are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy when indicated.
* Major surgery (under general anesthesia) within 30 days prior to therapy.
* Evidence of active hepatitis B virus (HBV) or hepatitis C virus (HCV), or history of HCV.
* Live vaccine within 30 days.
* Prior PD1, PD-L 1 or checkpoint inhibitors including CTLA4, Lag3, 41BB etc. within 2 years, or at any time if administered with the intent to treat Richter syndrome.
* Subjects with an active, known or suspected autoimmune disease. Subjects with type I diabetes mellitus, hypothyroidism only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis) not requiring systemic treatment, well controlled asthma and/or mild allergic rhinitis (seasonal allergies) are eligible.
* Evidence of central nervous system (CNS) involvement.
* Use of strong CYP3A4 inhibitors or inducers within 2 weeks prior to starting study therapy.
* History or concurrent condition of interstitial lung disease and/or severely impaired lung function.
* Patients with hemoglobin (Hb) A1c \> 8.5% at screening.
* Uncontrolled arterial hypertension despite optimal medical management (per investigator's assessment).
* Patients with uncontrolled coagulopathy or bleeding disorder.
* The following cardiovascular abnormalities:

* Congestive heart failure \>= class 3 New York Heart Association (NYHA) class.
* Unstable angina (angina symptoms at rest), new-onset angina (onset within the last 3 months).
* Myocardial infarction less than 6 months before start of study treatment.
* Left ventricular ejection fraction (LVEF) less than 45%.
* Corrected QT interval (QTc) \> 480 msec (if echocardiogram performed during screening).
* Females who are pregnant or nursing. Pregnant individuals are excluded from this study because copanlisib and nivolumab have the potential to cause fetal harm based on relevant animal studies (Refer to the appropriate prescribing information). Because there is an unknown but potential risk for adverse events in nursing infants, breast-/chest-feeding should be discontinued prior to treatment with copanlisib and nivolumab.