Burger lovers will have to get used to eating off fast-food trays inside McDonald’s again.

McDonald’s Corp. is facing tight supplies of some of its paper to-go bags, the latest supply challenge affecting U.S. restaurants. The chain recently told restaurant owners that they needed to limit orders of bags from suppliers as usage is running ahead of already high numbers last year, McDonald’s said in an internal message.

Many customers are asking for their Big Macs, McNuggets and fries in bags even when they dine inside, driving the tightness, the chain said in the message. Workers who have spent months packing all orders to go during the pandemic aren’t used to serving meals on trays, contributing to the strain, the company said.

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“Many new crew members have never had to deal with trays before,” the company said in the late-July message. The “transition to using trays has been slower, more difficult because we haven’t done it in so long.”

McDonald’s said that while bag supplies have been pressured for the chain and others in the industry, impact to its restaurants was minimal. “We are confident customers should not see disruption,” the company said. “We will continue to watch closely.”

Restaurant sales have risen from the depths of the pandemic, but operations are far from normal for even big chains like McDonald’s. Supply-chain hurdles remain, labor continues to be scarce and costs are rising. McDonald’s said prices at its U.S. restaurants are up around 6% from the same time last year as it handles labor and food cost pressures.

McDonald’s also recently told restaurants that they would need to limit their straw purchases.

Photo: Wilfredo Lee/Associated Press

At McDonald’s and other U.S. restaurants, to-go food sales remain elevated this year compared with pre-pandemic levels, even at sit-down establishments that did little takeout business before the crisis. To-go orders accounted for nearly 81% of restaurant business in the 12 months ended June 30, up 18% compared with the prior year’s period, according to consumer-research firm the NPD Group. Carryout, drive-through and delivery orders were all up, NPD data shows.

The Delta variant of the coronavirus is pushing more customers to order their food to go, with growth in fast-food takeaway sales in the week ended July 25 the strongest in the past six weeks, according to Black Box Intelligence, an industry analytics firm.

“It’s wreaking havoc everywhere,” said Susan Watkins, a 59-year-old Dallas area resident, about the Delta variant. Ms. Watkins, a cancer survivor, says she hasn’t been eating inside fast-food restaurants and gets her McDonald’s coffee from the drive-through.

The to-go boom is prompting a spike in demand for the paper that goes into takeaway bags. Shipments for unbleached bag and sack paper were up 12% last year compared with 2019 levels, according to the American Forest and Paper Association.

McDonald’s has brought back dine-in service as locations have lifted pandemic-related restrictions. Restaurants that have restored dine-in service are going through more bags than ones that haven’t, the company said. McDonald’s is instructing restaurants that dine-in orders should be placed on trays and encouraging operators to spread the word.

Jon Rathbun, a 48-year-old Little Falls, N.Y., resident, said he has noticed his local McDonald’s restocking trays lately, but said that many workers seem to prefer to put food in bags. Mr. Rathbun said he doesn’t like when restaurants use too many bags on orders because he views it as a waste.

Supply chain disruptions related to the coronavirus have hit other supplies, from ketchup to cups, restaurant companies said. Some Starbucks Corp. stores said they were running short of cups, plastic lid toppers and syrups earlier this year. The company recently said it has plenty of cups, but supplies are tight in other products, such as oat milk.

Mexican quick-service chain Del Taco Restaurants Inc. told investors late last month that it was trying to manage packaging shortages, which executives said stemmed from U.S. suppliers’ short-handed production plants.

Packaging costs are up 10% compared with last year for chain Noodles & Co., largely because of the raw material resin getting more expensive, Chief Executive Dave Boennighausen said. Power outages in Texas in February triggered chemical plant shutdowns, leading to a continuing shortage of resin and other materials used to make plastics.

Beyond paper bags, McDonald’s recently told restaurants that they would need to limit straw purchases, too. One of the chain’s suppliers in the U.S. is running short because of labor shortfalls, rail disruptions, West Coast wildfires and resin outages, the company said in the message to restaurant owners.

McDonald’s said it doesn’t expect to run out of straws, but it anticipates having to limit deliveries to restaurants for as many as two months. It is searching for new suppliers in the U.S. and Europe, the chain said.

—Robert Tita contributed to this article.

Write to Heather Haddon at heather.haddon@wsj.com