Huge queues formed leading to Terminals 2 and 3 as the motorway was shut in both directions between Junction 4 and Junction 4A while an inspection is carried out.
A free terminal transfer service is available from Terminals 4 and 5, Heathrow said.
One person claimed on Twitter to have been told it will take up to six hours to reopen the tunnel.
Cars have been unable to leave the airport’s gridlocked car park.
Buses are on diversion are the services - 105, 111, 285, 278, A10, X26, X140 and U3.
A spokesman for Heathrow said: “The tunnel to Terminals 2 and 3 is partly closed while our teams and the Fire Service carry out checks.
“There is congestion across Heathrow’s road network and passengers travelling to Terminals 2 and 3 are advised to avoid travelling by road and to instead use the rail network and underground.
“They may also use the free terminal transfer service from Terminals 4 and 5.
“If you are travelling by road, please allow extra time for your journey. We apologise for any inconvenience caused.”
A Metropolitan Police spokesman said: “Officers were called at 1.15pm on Tuesday, October 4 to the M4 near Heathrow airport. The East Bore and West Bore tunnels had triggered gas alarms.
“The roads have been closed as a precaution. LFB and Gas Board are attending.”