The last time U.S. oil prices were at $70, in 2014, they were in the middle of a steep collapse. Many investors believed then that prices would soon stabilize, or even recover. Instead, they continued to plunge, eventually hitting a bottom in 2016 at $26. That tumble caused acute pain for oil producers, whose troubles rippled out into stocks, bonds and the broader economy.

This year’s rally is a sign of how much has changed in a few years. Global growth has picked up, while U.S. unemployment has fallen. A gambit by the world’s largest oil producers to cut production has been succeeding in eliminating a massive glut, with help from soaring demand.

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Oil prices have climbed more than 60% since last summer’s lows, and U.S. producers are exporting more crude than ever.

For now, some investors say oil prices are lodged in a range that could benefit the U.S. economy by bolstering the recovering energy industry without curtailing demand.

Yet even with the economy chugging along, rising oil prices dredge up fresh concerns. If crude continues to move higher, it could begin to stifle economic growth. Higher consumer prices for gasoline and other energy products act like a tax, while pushing inflation higher and increasing pressure on the Federal Reserve to raise interest rates more aggressively.

That, in turn, could slow growth and weigh on the stock market, which has already been knocked around by trade tensions, rising bond yields and recent bouts of volatility. Inflation concerns pushed the yield on the 10-year Treasury note to the highest since 2014 on Friday, while major U.S. stock indexes closed lower, wiping out much of the recent gains after a string of upbeat earnings.

“Nothing can suck cash flow out of the economy faster than rising oil prices,” said Joseph LaVorgna, chief economist for the Americas at Natixis.

When oil prices fell below $40 a barrel, financial distress from the energy sector started to spread, said Jason Thomas, director of research at the Carlyle Group.

But if oil prices continue rising, they could boost inflation expectations, which would raise bond yields and the cost of financing.

“We’re starting to move out of that Goldilocks zone,” Mr. Thomas said. “Certainly $10 to $15 a barrel more there starts to be this drag.”

President Donald Trump tweeted Friday that oil prices are “artificially Very High!”—a sentiment that would have been unthinkable even a few months ago. Oil prices tumbled after his comment, but recovered to settle at $68.38 a barrel Friday.

A major force behind rising oil prices has been a policy reversal from the Organization of the Petroleum Exporting Countries. In 2014, the group opted to continue pumping oil at high rates in an effort to protect its market share against encroaching U.S. shale producers. Two years later, OPEC reversed course, enlisting other major producers such as Russia in a coordinated production cut that has helped to nearly eliminate a supply overhang.

“The conversation is changing,” said Antoine Halff, senior research scholar at Columbia University’s Center on Global Energy Policy. “A year ago the conversation was ‘lower for longer’ and the ‘age of abundance’” for oil, he said. Now, “the idea of cheap oil forever is being challenged.”

A booming global economy has also been key, keeping demand high as excess oil and fuel gets soaked up by consumers around the world. The first quarter was likely the strongest for global oil demand growth, year over year, since the fourth quarter of 2010, Goldman Sachs said.

But higher prices could threaten that. When drivers take to the road this summer, they will likely be paying the highest prices for gasoline since 2014. That will likely negate any financial benefits from tax cuts this year for low-income households, according to Deutsche Bank, and could further eat into disposable income.

“The higher prices get, over all, the consumer side of the economy will be affected. It’s like a tax increase on consumers as gasoline prices go up,” said Ann-Louise Hittle, vice president of oil markets at research firm Wood Mackenzie.

Some analysts believe that concerns that higher prices will cut into demand are overblown.

For one thing, oil’s gains have been gradual. Gasoline prices are still far from highs in 2008, when the national average topped $4 a gallon at times.

Demand has remained strong even as oil and fuel prices have been rising, and many analysts believe that prices still aren’t high enough to prompt big changes in behavior.

And with the U.S. now on track to overtake Russia as the world’s largest oil producer, a large swath of the U.S. economy stands to benefit from higher prices. Oil prices are even higher abroad, which has made it lucrative for U.S. producers to ship more crude overseas. Brent, the global benchmark, climbed to $74.06 a barrel Friday.

“In the past, any time oil prices have gone up it was as a result of supply constraints and the U.S. was at the mercy of foreign oil,” said Joseph Tanious, senior investment strategist at Bessemer Trust. “But U.S. oil production has picked up in a meaningful way—there could be also some benefits to having modestly rising oil prices.”

Oil’s rise has started to lift energy companies’ share prices, which had been slow to react to higher prices. Oil-and-gas companies have taken over as the U.S. stock market’s priciest segment, according to Credit Suisse analysts. Energy shares have gained 1.5% so far this year after a nearly 10% gain over the past month.

But even some producers worry about what will happen if higher oil prices stick around too long.

“We’re going to lose demand. It’s going to move more toward alternative energy,” Scott Sheffield, chairman of Pioneer Natural Resources Co., said at an energy conference last week. “I don’t think it does anybody any good to see $70, $80 crude.”