BGJ398 in Treating Patients With FGFR Positive Recurrent Head and Neck Cancer

Inclusion Criteria: * Histologically documented diagnosis of squamous cell carcinoma of the head/neck including nasopharyngeal carcinomas (lymphepithelioma histology is ok if criteria 2 is met) * Patients must have progressed on prior platinum based therapy (or have become intolerant) prior to enrollment on this study * Prior anti-PD-1 or other immunotherapy is acceptable * Known FGFR genetic alterations (specifically FGFR1-3 mutation, amplification, or translocation) via deoxyribonucleic acid (DNA) or ribonucleic acid (RNA) based assay. * The following genetic aberrations will be screened for: * FGFR1 amplification, FGFR1 somatic mutations, FGFR1 translocations * FGFR2 somatic mutations, FGFR2 translocations, FGFR2 amplification * FGFR3 somatic mutations, FGFR3 translocations, FGFR3 amplification * Other genetic FGF/FGFR pathway aberrations may be acceptable should such genetic changes be observed to emerge and require approval per the lead investigator for enrollment. * The number of enrolled patients with each type of genetic aberration may be limited at the discretion of the lead investigator. * Consent to undergo a fresh biopsy in case of benefit from therapy and subsequent progression * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 * Patients must provide written informed consent prior to any screening procedures * Aged 18 years or older * Willing and able to comply with scheduled visits, treatment plan and laboratory tests * Patient is able to swallow and retain oral medication, unless approval per the manufacturer of other administration routes/methods is provided * Recovery from adverse events of previous systemic anti-cancer therapies to baseline or grade 1, except for: * Alopecia * Stable neuropathy of =\< grade 2 due to prior cancer therapy * HPV status in oropharyngeal carcinomas; while HPV status (e.g. via p16) does not have to be known prior to consenting, the HPV status (e.g. using p16 immunohistochemistry \[IHC\]) needs to be established prior to start of therapy * Presence of measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Availability of tumor tissue (e.g. formalin-fixed, paraffin-embedded \[FFPE\]) for genomic profile (typically 12 unstained FFPE 5-10 micron slides, minimum of 10) Exclusion Criteria: * History of another primary malignancy except adequately treated in situ carcinoma of the cervix or non-melanoma carcinoma of the skin or any other curatively treated malignancy that has not been treated in the prior 3 months or expected to require treatment for recurrence during the course of the study * Patients with metastatic central nervous system (CNS) tumors are allowed provided that they are clinically stable for a period of 30 days prior to study entry and there is not a requirement for steroid (other than close to physiologic doses) or anti-convulsant therapy; patients with leptomeningeal involvement are excluded * Patients who received a prior selective FGFR inhibitor in the recurrent/metastatic disease setting; prior use of a multikinase inhibitor that includes anti-FGFR activity is acceptable after review by the lead investigator * History and/or current evidence of tissue calcification including, but not limited to, the soft tissue, kidneys, intestine, myocardium and lung with the exception of calcified lymph nodes and asymptomatic coronary calcification * Current evidence of corneal or retinal disorder/keratopathy including, but not limited to, bullous/band keratopathy, corneal abrasion, inflammation/ulceration, keratoconjuctivitis, confirmed by ophthalmologic examination * Impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of oral BGJ398 (e.g., ulcerative diseases, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, small bowel resection) * History and/or current evidence of endocrine alterations of calcium/phosphate homeostasis, e.g., parathyroid disorders, history of parathyroidectomy, tumor lysis, tumoral calcinosis, etc unless approval from lead investigator/ collaborator is obtained * Treatment with any of the following anti-cancer therapies prior to the first dose of BGJ398 within the stated timeframes * Cyclical chemotherapy (intravenous) within a period of 2 weeks unless there are ongoing side effects \> grade 2 * Biological therapy (including small molecules, and/or) within a period of time that is =\< 2 weeks prior to starting study drug unless there are ongoing side effects \> grade 2 * Any other investigational agents within a period =\< 2 weeks prior to starting study drug unless there are ongoing side effects \> grade 2 * Wide field radiotherapy (including radioisotopes) =\< 2 weeks prior to starting study drug unless there are ongoing side effects \> grade 2 * Patients who are currently receiving treatment with agents that are known strong inducers or inhibitors of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) are prohibited * Enzyme inducing anti-epileptic drugs * Consumption of grapefruit, grapefruit juice, pomegranates, star fruits, Seville oranges or products within 7 days prior to first dose * Use of medications that are known to prolong the QT interval and/or are associated with a risk of torsades de pointes 7 days prior to first dose * Use of amiodarone within 90 days prior to first dose * Use of medications that increase serum levels of phosphorus and/or calcium * Current use of therapeutic doses of warfarin sodium or any other coumadin-derivative anticoagulants; heparin and/or low molecular weight heparins or other anticoagulants are allowed * Insufficient bone marrow function * Absolute neutrophil count (ANC) \< 1,000/mm\^3 \[1.0 x 10\^9/L\] * Platelets \< 75,000/mm\^3 \[75 x 10\^9/L\] * Hemoglobin \< 10.0 g/dL * Insufficient hepatic and renal function * Total bilirubin \> 1.5 x upper limit of normal (ULN) (unless evidence of Gilbert's disease) * Aspartate aminotransferase (AST)/serum glutamic oxaloacetic transaminase (SGOT) and alanine aminotransferase (ALT)/serum glutamate pyruvate transaminase (SGPT) \> 2.5 x ULN * Serum creatinine \>= ULN and/or calculated or measured creatinine clearance \< 75% lower limit of normal (LLN) * Calcium-phosphate homeostasis * Inorganic phosphorus outside of normal limits * Total and ionized serum calcium outside of normal limits * Clinically significant cardiac disease including any of the following: * Congestive heart failure requiring treatment (New York Heart Association \[NYHA\] grade \>= 2) * Left ventricular ejection fraction (LVEF) \< 50% as determined by multigated acquisition (MUGA) scan or electrocardiogram (ECHO), or uncontrolled hypertension (refer to World Health Organization-International Society of Hypertension \[WHO-ISH\] guidelines) * History or presence of clinically significant ventricular arrhythmias, atrial fibrillation, resting bradycardia, or conduction abnormality * Unstable angina pectoris or acute myocardial infarction =\< 3 months prior to starting study drug * Corrected QT Interval Fridericia (QTcF) \> 450 msec (both genders) * History of congenital long QT syndrome * Pregnant or nursing (lactating) women * Known positive serology for human immunodeficiency virus (HIV), active hepatitis B, and/or active hepatitis C infection * Study medication cannot be administered through gastric (G)-tube, unless additional information from the manufacturer becomes available in the future * 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 3 months following the discontinuation of study treatment must be used by both sexes (= female patients and their male partners) * Post-menopausal women are allowed to participate in this study; women are considered post-menopausal and not of child bearing potential if they have had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral oophorectomy (with or without hysterectomy) or tubal ligation at least six weeks ago; in the case 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 males unless they use a condom during intercourse while taking drug and for 3 months after the last dose of the study drug and should not father a child in this period; a condom is required to be used also by vasectomized men in order to prevent delivery of the drug via seminal fluid