In my experience, these kinds of things aren't worth the money or effort. You'll see spikes in user activity when the contests are going on, then they'll disappear once they're over.
I find it's far more effective to create good and unique content that folks would be searching for, and run ad campaigns on that content. When folks find info, or discussions, that they care about, and get involved with, they'll want to keep coming back for more.
That said, I know lots of sites do these contests so maybe they do work for some. I'm just sharing my experiences.
I would say it depends on the community and the type of forum.
For a gaming forum around specific games things like concept art or similar contests can bring in a lot of activity and discussion. The same can go for other niche topics that get deeply passionate users.
For a general topic forum... The activity brought by a contest generally leads to low effort content. This is not always the case, but usually what I see.
The site I did contests regularly on saw a lot of good activity and user engagement, but also saw a lot of negative interactions due to people trying to rig or being hostile towards winners.
The shirts at Cafe Press that I had done (was looking at doing some more, but dang they've jumped up in price - now $33 each). The hats I'm still pricing around to find the best "bang for the buck". Most of them sit around $27 each.
Also looking at doing some mouse pads ($7.99 each), especially since so much of this field is done at computers.
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Printful (overall quality and selection is good; prices are a bit higher than others)
Printify (prices and quality are hit and miss, but a lot of options. Tools are good)
Merchize (good prices and selection, iffy tools/API support)
CustomCat (good prices and decent selection, iffy API support)
SPOD (good prices, decent selection, good tools, iffy API support)
All of these "work" with Wordpress, but to varying degrees. Printful works the best with WooCommerce, while the others have a few different issues either on the tool side or the exporting to WooCommerce side.