Lenalidomide and Temsirolimus in Treating Patients With Relapsed or Refractory Hodgkin Lymphoma or Non-Hodgkin Lymphoma

Inclusion Criteria: * Histology: bone marrow biopsies (with the exception of lymphoplasmacytic lymphoma) as the sole means of diagnosis are not acceptable; fine needle aspirates are not acceptable * Phase I: previously treated, histologically confirmed Hodgkin and non-Hodgkin lymphomas; the only exception to a requirement for a lymph node biopsy is lymphoplasmacytic lymphoma, which can be diagnosed based on morphologic evidence in the bone marrow plus the appropriate paraprotein * Phase II: previously treated, histologically confirmed mature non-Hodgkin lymphoma (NHL) stratified by histology: * Group A: diffuse large B-cell lymphoma (NOTE: all patients with DLBCL must have germinal center vs. non-germinal center phenotype established via immunohistochemistry) * Group B: follicular lymphoma * Group C: lymphoma NOS (including Hodgkin lymphoma, T-NHL, marginal zone lymphoma, lymphoplasmacytic * No limit to number of prior therapies; prior autologous transplantation is allowed * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Absolute neutrophil count (ANC) \>= 1,000/uL * Platelet count \>= 75,000/uL * Total bilirubin =\< 1.5 times upper limit of normal (ULN) (unless due to Gilbert's) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 times ULN * Creatinine clearance \>= 60 mL/min as determined by calculated Cockcroft-Gault equation * Fasting serum cholesterol =\< 350 mg/dL * Fasting serum triglycerides =\< 2.5 times ULN * All patients are required to have measurable disease; non-measurable disease alone is not acceptable; any tumor mass \> 1 cm is acceptable; lesions that are considered non-measurable include the following: * Bone lesions (lesions if present should be noted) * Ascites * Pleural/pericardial effusion * Lymphangitis cutis/pulmonis * Bone marrow (involvement by lymphoma should be noted) * For Waldenstrom's macroglobulinemia, measurable disease is defined as at least one lesion with a single diameter of greater than 2 cm by computed tomography or bone marrow involvement with greater than 10% malignant cells and quantitative monoclonal protein (immunoglobulin M \[IgM\], IgG, IgA) greater than 1,000 mg/dL * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 10 - 14 days and again within 24 hours prior to starting cycle 1 of lenalidomide; further, they must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control: one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before starting lenalidomide; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP, even if they have had a successful vasectomy; a FCBP is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months); all patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * Ability to understand and the willingness to sign a written informed consent document * Patients who are human immunodeficiency virus (HIV) positive are allowed to participate BUT must meet the following criteria: * No acquired immune deficiency syndrome (AIDS)-defining illness, AND * Cluster of differentiation (CD) 4 count \>= 400 cells/mm\^3, AND * No anti-retroviral therapy (including high-active antiretroviral therapy \[HAART\]) within 7 days of starting protocol therapy, AND * Patient may not take concurrent anti-retroviral therapy (including HAART) while on protocol * NOTE: it is not generally recommended to suspend anti-retroviral therapy (including HAART); the medical team enrolling a patient who suspends anti-retroviral therapy for the purpose of study participation must have a documented note reviewing the potential risks/benefits with the patient in the medical chart Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Patients who are receiving any other investigational agents * Patients with known brain metastases should be excluded from this clinical trial * History of allergic reactions attributed to compounds of similar chemical or biological composition to temsirolimus or lenalidomide used in study * Patients requiring active anti-retroviral therapy for HIV are excluded * No "currently active" second malignancy, other than non-melanoma skin cancers; patients are not considered to have a "currently active" second malignancy if they have completed anti-cancer therapy and are considered by their physicians to be at less than 30% risk of relapse * No history (within 3 months of study entry) of deep venous thrombosis/pulmonary embolism (DVT/PE); patients with a distant history (greater than 3 months before study entry) of DVT/PE are eligible, but should receive prophylactic aspirin or low molecular weight heparin * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Patients with relapsed/refractory DLBCL or HL who are eligible and willing to undergo potentially curative stem cell transplant * Patients with chronic lymphocytic leukemia (CLL)/small lymphocytic lymphoma (SLL) are excluded * No corticosteroids within 14 days prior to study, except for maintenance therapy for a non-malignant disease; maintenance therapy dose may not exceed 10 mg/day prednisone or equivalent; any patient on steroid therapy must receive thromboembolic prophylaxis