Temsirolimus With or Without Cetuximab in Patients With Recurrent and/or Metastatic Head and Neck Cancer Who Did Not Respond to Previous Therapy

Inclusion Criteria: * Histologically/cytologically confirmed diagnosis of squamous cell carcinoma of head and neck origin not amenable to curative intent therapy; information on prior exposure to cetuximab (duration, single agent/combined with chemotherapy/combined with radiation, best response, interval prior to study entry) will be collected * Progressive disease by RECIST criteria (or unequivocal clinical progression) on a cetuximab based therapy in any line of therapy for recurrent/metastatic disease; prior use of cetuximab for recurrent/metastatic disease is defined as palliative intent use either alone or in combination with chemotherapy with a minimum of 2 weeks of uninterrupted treatment with cetuximab; treatment with cetuximab during radiotherapy or chemoradiotherapy is not sufficient * Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 * Presence of measurable lesions by RECIST: patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan * Knowledge of the anatomic site of the original tumor (oropharynx versus non-oropharynx) or alternatively human papilloma virus (HPV) status; the trial will stratify patients by oropharynx versus non-oropharynx origin; HPV(+) tumors will be counted in the oropharynx cohort, HPV(-) tumors in the non-oropharynx cohort; at a later point all patients will undergo HPV testing as part of this trial; any widely used form of HPV testing is acceptable (including but not limited to HPV in situ hybridization \[ISH\], p16 testing \[immunohistochemistry (IHC)\], HPV16 testing, polymerase chain reaction \[PCR\], hybrid capture, etc) * Availability of formalin-fixed, paraffin-embedded (FFPE) tissue and blood * FFPE: \>=14 slides containing tumor, 18 recommended * 7-10 slides 5 um thick, AND 7-10 slides 10 um thick, and cut with a clean blade (use new blade if possible or clean vigorously to avoid RNA/DNA, RNase contamination) * Blood: two 10 cc ethylenediaminetetraacetic acid (EDTA) purple top tubes (blood); two 2 ml cryovials (serum) * Patients with human immunodeficiency virus (HIV), not requiring highly active antiretroviral treatment (HAART) therapy are eligible * Life expectancy of greater than 8 weeks * Leukocytes \>= 2,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits (unless proven Gilbert's disease, which after principal investigator \[PI\] approval patient may be included) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal * Creatinine within 1.5 X normal institutional limits * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation * Ability to understand and the willingness to sign a written informed consent document * Fasting glucose of =\< 120 mg/dl and glycosylated hemoglobin (HbA1c) =\< 7.5% Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 2 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 2 weeks earlier * Patients may not be receiving any other investigational agents * Patients with known, active brain metastases should be excluded from this clinical trial; patients with treated brain metastases stable for \>= 12 weeks are eligible * History of allergic reactions attributed to compounds of similar chemical or biologic composition to temsirolimus or cetuximab * Concurrent life-threatening diseases: patients with diseases which with reasonable certainty do not limit life expectancy to 12 months or less are eligible; assessment of such concurrent illnesses should be by the principal investigator * Use of strong inhibitors/inducers of CYP3A4 is not permitted * 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 * Pregnant women are excluded from this study * Breastfeeding should be discontinued if the mother is treated with temsirolimus * HIV-positive patients with normal immune function (CD4 count \> 200) are eligible if there are no drug interactions with temsirolimus or cetuximab; patients with impaired immune function are ineligible due to the risk of additional immunosuppression from temsirolimus therapy * Concurrent administration of temsirolimus with vaccinations is to be avoided and a 14-day window from administration of the vaccine is advised; in emergent situations this policy may be revisited by the PI if deemed important for the patient's health * Poorly controlled hyperglycemia (HbA1C \> 7.5%) or hyperlipidemia are exclusion criteria; hyperglycemia or hyperlipidemia need to be appropriately managed and controlled * Concurrent use of warfarin is allowed, but requires close monitoring of prothrombin time (PT)/international normalized ratio (INR) * Patients with clinically significant pneumonitis/pulmonary infiltrates unless there is a known and treatable cause for the condition