This paper reports the results of a near-infrared spectroscopic survey of LINER galaxies undertaken with a new infrared spectrograph at the 5 m Hale telescope. The galaxy sample includes 11 LINERs with spectra covering the [Fe II] (1.2567 mu m), Pa beta (1.2818 mu m), H-2 [1-0 S(1), 2.1218 mu m], and Bry (2.1655 mu m) near-infrared emission lines and one additional galaxy with only [Fe II] and Pa beta line coverage. All of the LINERs with infrared line detections have strong [Fe II] and/or H-2 emission, with about half (four of nine) having extremely high ratios (>2) of [Fe II] to Pa beta. The strength of the H-2 and [Fe II] lines is well correlated with the optical [O I] line, with many LINERs having higher ratios of [Fe II]/Pa beta, H-2/Bry, and [O I]/H alpha than other galaxy types. The LINERs with the highest [Fe II]/Pa beta ratios (termed "strong" [Fe II] LINERs) show evidence for recent star formation. Shocks from compact supernova remnants may enhance the [Fe II] emission in these "strong" [Fe II] LINERs. The LINERs with lower [Fe II]/Pa beta ratios (termed "weak" [Fe II] LINERs) are more consistent with Seyfert-like activity, including higher ionization states, some strong X-ray sources, and some broad Ha detections. The [Fe II] luminosity and the [Fe II]/Pa beta ratio in these objects are more easily explained by hard X-ray excitation than in the "strong" [Fe II] LINERs. These "weak" [Fe II] LINERs are considered prime candidates for being low-luminosity Seyfert nuclei.