  Early in their lives, planets endure extreme amounts of ionizing radiation
from their host stars. For planets with primordial hydrogen and helium-rich
envelopes, this can lead to substantial mass loss. Direct observations of
atmospheric escape in young planetary systems can help elucidate this critical
stage of planetary evolution. In this work, we search for metastable helium
absorption---a tracer of tenuous gas in escaping atmospheres---during transits
of three planets orbiting the young solar analogue V1298 Tau. We characterize
the stellar helium line using HET/HPF, and find that it evolves substantially
on timescales of days to months. The line is stable on hour-long timescales
except for one set of spectra taken during the decay phase of a stellar flare,
where absoprtion increased with time. Utilizing a beam-shaping diffuser and a
narrowband filter centered on the helium feature, we observe four transits with
Palomar/WIRC: two partial transits of planet d ($P = 12.4$ days), one partial
transit of planet b ($P = 24.1$ days), and one full transit of planet c ($P =
8.2$ days). We do not detect the transit of planet c, and we find no evidence
of excess absorption for planet b, with $\Delta R_\mathrm{b}/R_\star<0.019$ in
our bandpass. We find a tentative absorption signal for planet d with $\Delta
R_\mathrm{d}/R_\star = 0.0205\pm0.054$, but the best-fit model requires a
substantial (-100$\pm$14 min) transit-timing offset on a two-month timescale.
Nevertheless, our data suggest that V1298 Tau d may have a high present-day
mass-loss rate, making it a priority target for follow-up observations.
