Some more details about the situation would be nice. Do players ever die this way, or does it only occur with bots? Does the door sometimes close even while the player/bot is in the trigger volume? Are there any errors in your compile log or alt-p problem checker? Etc.
Here are a couple possibilities to try:
-Could it just be the TFBots changing their class? At the start of a round, TFBots may suicide a couple times as they pick a class to ensure a proper team composition. It shouldn't take bots too long to settle on a class, only a few seconds really, and they should respawn instantly after suiciding. If they aren't respawning instantly, check that your func_respawnroom is set up properly by changing class yourself. You can test with the command "tf_bot_reevaluate_class_in_spawnroom 0" to get bots to stop attempting to change class by suiciding. If bots continue dying, you have a different problem.
-Are they being crushed by the spawnroom door as it closes? If your door is closing and reopening while players/bots are in it, your door logic isn't set up right. One possible cause is not setting "Delay Before Reset" to -1 in your func_door properties. If your door is set to damage blocking entities, it will kill anyone unlucky enough to be in the door's path when it closes.
-Does your map use any trigger_hurt entities that cover sections of the map near where the bots are dying? These could be enabling at inopportune times, killing bots.
Also, be warned that bots are dumb and unhumanlike, and as such any map designed around bot gameplay will play poorly with humans (unless it is MvM). It's fine if you want to just screw around or test map logic, but designing and balancing your map's layout around how it plays with bots likely won't go well. If you want to test with humans, submit it to an IMP test on the TF2Maps servers. Not only will you get to play it with real live human players, but players can leave feedback on the map that you can later view, and you can find a demo file which allows you to rewatch the match as if you were spectating it.