The SPitzer InfraRed Intensive Transients Survey (SPIRITS) searched for luminous infrared (IR) transients and variables in nearly 200 nearby galaxies from 2014 to 2019, using the warm Spitzer telescope at 3.6 and 4.5{mu}m. Among the SPIRITS variables are IR-bright objects that are undetected in ground-based optical surveys. We classify them as (1) transients, (2) periodic variables, and (3) irregular variables. The transients include eSPecially Red Intermediate-luminosity Transient Events (SPRITEs), having maximum luminosities fainter than supernovae, red IR colors, and a wide range of outburst durations (days to years). Here we report deep optical and near-IR imaging with the Hubble Space Telescope (HST) of 21 SPIRITS variables. They were initially considered SPRITE transients, but many eventually proved instead to be periodic or irregular variables as more data were collected. HST images show most of these cool and dusty variables are associated with star-forming regions in late-type galaxies, implying an origin in massive stars. Two SPRITEs lacked optical progenitors in deep preoutburst HST images; however, one was detected during eruption at J and H, indicating a dusty object with an effective temperature of ~1050K. One faint SPRITE turned out to be a dusty classical nova. About half the HST targets proved to be periodic variables, with pulsation periods of 670-2160d; they are likely dusty asymptotic-giant-branch (AGB) stars with masses of ~5-10M_{sun}_. A few of them were warm enough to be detected in deep HST frames, but most are too cool. Out of six irregular variables, two were red supergiants with optical counterparts in HST images; four were too enshrouded for HST detection.