Will I go through customs at Heathrow when traveling from Dulles via Heathrow (airline change from United to TAP) to Lisbon?
Short answer - no, you won't have to clear customs
Long answer - that doesn't however mean it'll be as simple as changing between two domestic flights in the USA though...
Both United and TAP are part of Star Alliance. As such, you'll be able to check your luggage through between the two flights. Depending on the checkin agent, it may not be a 30 second job, but as long as you have the details of both flights to hand (including both PNR and e-ticket numbers), you'll be able to check your bags through to the final destination
When you arrive in Heathrow, with both airlines being Star Alliance, you'll land in Heathrow Terminal 2, the same terminal as departure. You'll follow the Purple Flight Connections signs, which'll take you to the main bit of the terminal (T2A). Here, you'll go through security, but not immigration, nor customs.
After clearing the transit / transfer security (normally quite quick, unless at a busy time), you'll then be in the main terminal area. If you don't have a boarding pass already, there are airline transit desks in the connections security area who can issue you with one. Once you have your boarding pass and are through security, check the monitors for where you flight goes from. Most, but not all short haul go from T2A, which is where you'll be. Most, but not all long haul go from T2B, which is a short escalator ride + walk + escalator ride away. Check the monitors, they'll tell you where to go!
For some reason, if it were a different airline combination not in star alliance, you might have to go through immigration and customs to get the bag to re-check. However, with both TAP and United being Star Alliance, you can check all the way through.
TL;DR - Immigration, no. Customs, no. Security, yes