You cannot stay airside at Heathrow Airport overnight. You can however remain in the public areas and try and sleep (I wouldn't, as I would worry about my belongings getting stolen). There are no sleeping couches, just ordinary seating. If you are worried about visa requirements though, you may be exempt.

While you can technically stay in the public "landside" areas of London Heathrow (LHR) overnight, it is not a comfortable experience and is generally discouraged unless you have an early morning flight. The airport does not close, but the "airside" departure lounges are cleared of passengers after the last flight (usually around midnight) and don't reopen until 5:00 AM. If you are stranded, you will have to stay in the Arrivals area, which has limited seating and can be quite cold and bright. For a much better experience, Heathrow features several on-site hotels like the Aerotel in Terminal 3 or the YOTELAIR in Terminal 4, which offer "day rooms" or overnight stays directly inside the terminal. There are also several high-end hotels like the Sofitel (T5) and Hilton (T4) connected via covered walkways. If you are on a budget, taking the Elizabeth Line or a shuttle to the "Bath Road" hotel strip offers much cheaper accommodation than staying in the terminal, where security may periodically ask to see your boarding pass to ensure you have a legitimate reason for being there.