Is 1 Corinthians 10:24 telling Christians to try to get what other people have?
No. The verse does not instruct Christians to covet or acquire what others possess. Paul is not saying you are to try to get what the next man has, since that contradicts everything he teaches. Instead, believers should not focus on what benefits themselves, but on what enhances their neighbor's situation.
Les explains that the italicized word "wealth" in many translations is unfortunate and should have been left blank. He stresses that Paul is not telling believers to try to get what the next man has, because that would contradict everything Paul teaches. Rather, Paul is saying we should not constantly look for what is in it for number one—"what's in it for me?"—but should be concerned about what our actions do for our neighbor and how they enhance his situation. That is exactly what Paul is driving at in this verse.
Let no man seek his own, but every man another’s wealth.