There are several sites which feature catalogs of cameras, searchable by feature. Unlike a Stack Exchange question with answers listing current possibilities, these databases are kept up to date by design. Unfortunately, the one at Neocamera doesn't appear to have "has remote" as a search option, but Digital Photography Review does — see this search result list. They don't have a way to narrow by price, but that's not too much of a hardship as they do list estimated current prices (they're owned by Amazon, so... that's easy) and you can just go through until you find a hit.

Perhaps more importantly for your case, though, they don't differentiate by IR vs. other technology — my new Fujifilm camera uses Wifi plus a smartphone app. For many people, the exact technology involved is not such a big deal, but from this comment thread it looks like you're looking specifically for something to work with the Foldio360. In that case, as @scottbb notes, the Kickstarter says they support Canon, Nikon, Pentax, and Sony — I think probably they support anything that does IR triggering on a similar frequency, but... not necessarily. I'd check first. Or, narrow your search to just these brands, and rule out the ones that are "Yes (via smartphone)".

Your budget is quite low, so I think you might be best served starting with older cameras and then searching for them used.