Can a player in a lawn tennis match request that the umpire be changed?

The supervisor of an ATP Tournament can do that. Here is an excerpt of the rulebook.

Section: V. Personnel / 5.06 Supervisor / D. Responsibilities

In all ATP Tour and ATP Challenger Tour tournaments the Supervisor shall:

4) Direct, supervise, instruct and evaluate the Referee, Chief of Umpires and all oncourt officials, including the authority to:

b) Remove a Chair Umpire and/or remove, rotate or replace any Line Umpire or net judge when necessary to improve the officiating of a match.

Moreover, the chair umpire has the right to remove a line umpire.

Section: V. Personnel / 5.07 Chair Umpire / Review Official / D. Responsibilities

In all ATP Tour and ATP Challenger Tour tournaments the Supervisor shall:

15) Remove, rotate or replace any Line Umpire or net judge whenever, in the opinion of the Chair Umpire, it will improve the officiating of a match.

However, this is an extremely rare case. I have never witnessed it and I watch a lot of tennis. Maybe it can happen more 'frequently' at the lower levels of professional competition (ATP Challenger Tour and the ITF Tour). Then again at these lower levels, there are often only 2 (if at all) line umpires on court. Removing one of them may make things even worse.

In the second round match between McEnroe and Nastase at the 1979 US Open it seems that the chair umpire was replaced. Read point 2 at this story. See a short documentary on this encounter here.

The rulebook can be found here: https://www.atptour.com/en/corporate/rulebook