  We present the results of a search for 10--1,000 GeV neutrinos from 2,268
gamma-ray bursts over 8 years of IceCube-DeepCore data. This work probes burst
physics below the photosphere where electromagnetic radiation cannot escape.
Neutrinos of tens of GeVs are predicted in sub-photospheric collision of free
streaming neutrons with bulk-jet protons. In a first analysis, we searched for
the most significant neutrino-GRB coincidence using six overlapping time
windows centered on the prompt phase of each GRB. In a second analysis, we
conducted a search for a group of GRBs, each individually too weak to be
detectable, but potentially significant when combined. No evidence of neutrino
emission is found for either analysis. The most significant neutrino
coincidence is for Fermi-GBM GRB bn 140807500, with a p-value of 0.097
corrected for all trials. The binomial test used to search for a group of GRBs
had a p-value of 0.65 after all trial corrections. The binomial test found a
group consisting only of GRB bn 140807500 and no additional GRBs. The neutrino
limits of this work complement those obtained by IceCube at TeV to PeV
energies. We compare our findings for the large set of GRBs as well as GRB
221009A to the sub-photospheric neutron-proton collision model and find that
GRB 221009A provides the most constraining limit on baryon loading. For a jet
Lorentz factor of 300 (800), the baryon loading on GRB 221009A is lower than
3.85 (2.13) at a 90% confidence level.
