I suspect the police might well move you on if you attempt to crash out...
Maybe, maybe not. You can certainly expect to be woken at intervals during the night and asked the reason for your presence, though you may well find that possession of a ticket/boarding card for an 07:00 departure is enough to stop you being turfed out into the cold night air.
The Heathrow website isn't a lot of help. All it says is
There is nowhere to sleep overnight in the [sic] terminal but there's a great choice of hotels nearby