Since Arkham Horror doesn't have a Rules forum yet I hope this is the right place to ask.
"Burglary: -->: Exhaust Burglary: Investigate. If you succeed, instead of discovering clues, gain 3 resources."
The question is, can you use Burglary to try and gain 3 resources in a room that has no clues, such as the Hallyway? Can you take an Investigate action in a room with no clues?
The arrow on Burglary indicates that it is a Triggered Ability. In the RRG, it says:
"A triggered ability can only be initiated if its effect has the potential to change the game state, and its cost (if any) has the potential to be paid in full, taking active cost modifiers into account. This potential is assessed without taking into account the consequences of the cost payment or any other ability interactions."
Based on this passage I'm tempted to say that no you cannot, as there is no clues so a successful Investigation would not change the game state.