Due to the flooding caused by hurricane Harvey, United is currently waiving change fees and any difference in fare if you rebook flights departing from, arriving at or connecting through IAH.
Doing this by phone is practically impossible: the waits were long this morning but now they are simply not accepting calls: I wuold like to connect you to an agent, but due to high call volume all agents are currently busy. Please call us back at a later time.
Up front, they suggest doing this online at united.com, but what isn't clear if flights where at least some leg is operated by a partner airline are eligible. Replacement flights for a $500 fare are close to $3,000 so I won't want to screw this up. United shows these results when you choose "change flights" but under the category "Partner flights".
An example route is operated variously by Copa Airlines, Air Canada, etc but has all UA flight numbers on the booking screen (but not on the later confirmation screen). They ask for my credit card at the end of the process.
For what it's worth, I went through the with rebooking process on united.com, and at the end got the oddly-capitalized error:
! united.com is unable to initiate your flight change request at this time. Please contact United Reservations at 1-800-UNITED1 to make changes to this Itinerary.
They still charged my credit card for the full difference between the two fights.