We present the results of a search for 10–1000 GeV neutrinos from 2268 gamma-ray bursts (GRBs) over 8 yr of IceCube-DeepCore data. This work probes burst physics below the photosphere where electromagnetic radiation cannot escape. Neutrinos of tens of giga electronvolts 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 tera electronvolt to peta electronvolt 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.