The answer depends on 4 things:
- How you are getting to the airport
- What airline you're flying with
- How frequent the flights on that route are
- How good your travel insurance is
If you have great travel insurance, are on a route with one flight roughly ever hour, and are just hopping on the tube for 20 minutes to get there, then arriving just before you're airline's cut-off time is probably fine.
If you're taking two trains a bus on a 3+ hour journey, with travel insurance that isn't very good about missed departure compensation, to take a once-a-day flight, I'd say aim to be there at least 3 hours before
The close of baggage check and at-desk checkin times varies between airlines. BA, for example, have a 45 minute cut-off at T3 and T5, AA it's 60 minutes, so check with your specific airline to see what they require.
Security queues can be bad at Heathrow, but not as bad as you find at some other airports world-wide. I often grumble about the Heathrow ones, but frankly I've had worse at JFK and Seattle recently! If you're a frequent traveller who knows how to properly unpack your bag, take out liquids properly, already has liquids in a suitable bag etc, you should be fine within the minimum checkin times. For less frequent travellers, I'd say 1 hour should be OK.
The question then is more about possible delays getting to the airport. You need to leave enough time so that if you get delayed, you won't miss the flight. The 2-3 hour figure tends to be more about that, at least for Heathrow. Think about what could go wrong, what's likely to go wrong, backup options, delay lengths etc. Then, check your travel insurance, and see what they require. If your travel insurance won't cover missed departures due to delays getting to the airport, go for 3 hours minimum unless really close. If your travel insurance says you must have aimed for 2.5 hours before if they'll cover it, obviously leave 2.5 hours!
Finally, there are shops airside in T3, and some restaurants, so it's not the end of the world if you get there with lots of time to spare. There are also some lounges you can pay to enter (the No.1 Traveller lounge in T3 is very nice, for example). If you have status or are travelling in a premium cabin, that could be a reason to get there extra early so you can enjoy the lounge more!
There is no universal answer