DoorDash Inc. will start delivering from McDonald’s Corp. restaurants in Houston later this month, a blow to Uber Technologies Inc.’s relationship with the burger giant as the only company currently ferrying its food to customers in the U.S.

The move comes as companies including Uber’s Uber Eats jockey for share in the fast-growing business to deliver food to customers’ homes and offices. Investors have poured money into startups including DoorDash and a rival, Postmates Inc., which filed earlier this year to list its shares...