We present a pilot search of CO emission in three H$_2$-absorbing, long-duration gamma-ray burst (GRB) host galaxies at z~2-3. We used the Atacama Large Millimeter/sub-millimeter Array (ALMA) to target the CO(3-2) emission line and report non-detections for all three hosts. These are used to place limits on the host molecular gas masses, assuming a metallicity-dependent CO-to-H$_2$ conversion factor ($alpha_{rm CO}$). We find, $M_{rm mol} < 3.5times 10^{10},M_{odot}$ (GRB,080607), $M_{rm mol} < 4.7times 10^{11},M_{odot}$ (GRB,120815A), and $M_{rm mol} < 8.9times 10^{11},M_{odot}$ (GRB,181020A). The high limits on the molecular gas mass for the latter two cases are a consequence of their low stellar masses $M_star$ ($M_star lesssim 10^{8},M_{odot}$) and low gas-phase metallicities ($Zsim 0.03,Z_{odot}$). The limit on the $M_{rm mol}/M_star$ ratio derived for GRB,080607, however, is consistent with the average population of star-forming galaxies at similar redshifts and stellar masses. We discuss the broader implications for a metallicity-dependent CO-to-H$_2$ conversion factor, and demonstrate that the canonical Galactic $alpha_{rm CO}$, will severely underestimate the actual molecular gas mass for all galaxies at $z>1$ with $M_star < 10^{10},M_odot$. To better quantify this we develop a simple approach to estimate the relevant $alpha_{rm CO}$ factor based only on the redshift and stellar mass of individual galaxies. The elevated conversion factors will make these galaxies appear CO-dark and difficult to detect in emission, as is the case for the majority of GRB hosts. GRB spectroscopy thus offers a complementary approach to identify low-metallicity, star-forming galaxies with abundant molecular gas reservoirs at high redshifts that are otherwise missed by current ALMA surveys.
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