If a grad student is working as a TA, can they also be asked to work as their adviser's lab manager for free?
I'm going to skip the legal question, because legal questions are always location specific (and you don't mention location), and I am not a lawyer (nor will you find many, if any, here). Instead, the question is whether you should put up with these circumstances.

Training/advising are probably worthwhile things to learn, and one could argue that those activities contribute to your education even as you teach others; upon graduation you can advertise your mentorship experience. Even better if you can train/advise others while they work on projects within or related to your own research interests. I don't consider these to be "lab manager" activities, though if it is taking up all your time that is of course a problem. As in all things, balance is important.

However: processing samples for others? With your lab as 'service center'? Unless this is a collaboration where you are getting authorship, this is a rare case where someone posts here thinking they are being abused that I think you actually are. Don't walk, RUN, away from this arrangement (note: this is a quote from regular Academia.SE contributor @JeffE as advice to students in bad situations and comes with important caveats).

Except probably first talk to your advisor and try to see if they can be reasonable. If your lab is providing a service, your lab should be compensated by those using the service. If you are doing the work of providing the service, that compensation should include you in some way.

To reiterate: there may be legal issues here, but you should start by having a conversation with your advisor about how to continue in a way that is mutually beneficial. If your lab has no funding, your advisor may not have many other choices and this may be the penalty you (both) pay for staying in their lab; they probably aren't happy about this either. If, on the other hand, they are using your availability to keep themselves afloat, that puts you in a strong negotiating position and you should use that position to come to an agreement that keeps your research moving forward.