Since the question assumes that the incumbent has lost, the talk of court challenges and the like is not relevant. If such a challenges were to succeed, the incumbent would be the winner.
Once Congress has finished with its 12th-amendment duties in determining the winner of the election, the result is decided. It is not necessary for defeated candidates to concede. Courts are generally unwilling to review political matters such as this.
If Congress determined that the incumbent president had lost the election, then, as a matter of law, that person would cease to be president at noon on January 20th. Refusing to leave the White House doesn't change that, and the new president's inability to enter the White House would not deprive him or her of the office of president.
By remaining in the White House without authorization, the outgoing president would be liable for conviction under 18 USC 1752 and a fine of up to $100,000 and/or imprisonment for up to one year (or a fine of up to $250,000 and imprisonment for up to 10 years if they resisted with weapons or if the resistance cause serious injury).
Is there any precedent for such an event? Are there any established processes in place to deal with such a situation?
As others have noted, there is certainly no precedent for this. I suppose the Secret Service probably has some procedure for this, but if they do it is probably, well, secret. In practice, as others have noted, the incoming president would likely decide what to do based on political considerations.