Yes, a derailleur can lose tension if a spring breaks. Note that this is an all or nothing proposition. Springs don't get weaker over time, but they do break eventually.
I don't know which spring you're referring to, so you test the idler cage spring by pushing the lower pulley forward to slacken the chain, then letting go and seeing if it takes up the slack.
The in/out return spring is harder to test because cable friction can be affecting it. Shift the bike to high gear while pedaling, and hopefully the RD will spring to the outside. If not, help by pulling it out while pedaling. Now disregard the levers and, while pedaling, shift to a lower gear by pushing the lower body in and then letting go and seeing if it moves back out. If it doesn't, the spring is dead & you need a new derailleur. If it does spring out this way, but not in normal use then the derailleur is fine, and your problem is cable friction, or possibly, but not likely the lever.
As I said, derailleurs don't wear out by losing spring tension, the wear would be in the pivots making for sloppy shifting, but the springs themselves are either as good as new, or dead.
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