Traps are typically not set off via failed Investigation checks
As iaminsensible's answer points out, typically failed Investigation checks to find traps in areas with traps simply result in no traps being found, as though you had rolled to Investigate an area that really didn't have any traps.
However, there has been a precedent set for failed attempts to disarm traps resulting in activating the trap, usually if you roll too low (say, the DC is 15, and you roll a total of below 10).
This option is also presented in Xanathar's Guide to Everything; some sample traps on page 114 includes the Crossbow Trap, which has the following:
Countermeasures. A successful DC 15 Wisdom (Perception) check reveals the trip wire. A successful DC 15 Dexterity check using thieves' tools disables the trip wire, and a check with a total of 5 or lower triggers the trap.
This is also further described on page 117 under Disarming a Simple Trap (the last paragraph).
This suggests that if anything was going to trigger a trap as a result of rolling too low, it would more likely be an attempt to disable it rather than to find it through Investigation (or Perception, alternatively). However, ultimately, the consequence of a failed skill check is up to the DM.