The hatch was left open for a number of reasons.
The Lunar Module is crowded with two people in it; with three, there's barely room to move. For much of the time, the crew would be spread between the LM and Command Module to keep from getting in each other's way. In particular, the CM was used as a sleeping area (the "upstairs bedroom"), since the active radio was in the LM.
One of the methods of dealing with CO2 buildup was to run the LM's suit supply hoses up the connecting passage to blow air into the CM. Even without active circulation, the lithium hydroxide canisters in the CM can absorb some CO2.
All electrical connections between the Command Module and the Lunar Module pass through the passageway between them. Normally, this is used to power the LM's systems from the Service Module's fuel cells, but it can be run in reverse to power the CM systems from the LM's batteries. Leaving the hatch open and the umbilical connected permits powering up CM systems if needed without draining the CM's batteries.
All information from the Apollo 13 Lunar Flight Journal.