  SPIRITS--the SPitzer InfraRed Intensive Transients Survey--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 microns.
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 "SPRITE"s
(eSPecially Red Intermediate-luminosity Transient Events), 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 pre-outburst HST images; however, one was
detected during eruption at J and H, indicating a dusty object with an
effective temperature of ~1050 K. 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 670-2160 days; they are likely dusty
asymptotic-giant-branch (AGB) stars with masses of ~5-10 Msun. 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.
