If a big company uses an open source library there's no contractual agreement that the developer must slave away for free and fix stuff for the big company. The developer, or the company that develops open source, can just essentially say "fuck it!" and do whatever they want.
Instead, the big company can take what's available. If it's not enough, they can contribute to the open source project (a wise move!), find a replacement project to ride on, or buy or write their own implementation.
Not all open source developers work for free but many do because they want to spend some of their time building something they care about. If the project has big impact sometimes the developers get hired to continue working on the software on a payroll. This is probably a win-win: the developer gets paid to do what he would be doing anyway and the company gets stability into future development.
If open source developers want to build a good, trusted brand of their software then it of course takes continuous involvement to support their users and can lead to burn-outs but that has nothing to do with open source. The exact same applies to building an established brand out of proprietary software.
And that effort only makes sense in the first place if there's a chance for a significant financial payout in the end, or the developers really, really just want fame in which case they have their own priorities on how they want to resource the development.
Still, nothing to do with open source per se.
I bet a thousand-fold more developers are burning out in jobs working on proprietary software.