The site I want to use says that it stores permanent cookies on my computer to track various things, for instance if I have two accounts with them. It says "Please note that if you set your browser to disable cookies, you may not be able to access certain parts of the Site for example applying for a job or posting a job. Other parts of the Site may also not work properly." Can I use it with TOR and will it get my real IP because of these persistent cookies?
Not inherently, no.
Cookies and IP addresses are independent. However, if you use the same browser for Tor and non-Tor access to the site, the same cookies will be sent both times, negating any benefit you would otherwise gain from using Tor.
Tor is not a cure-all for anonymity issues, it's merely one tool to use. There are many ways you can be tracked other than through your IP address, both technical (eg. logging in to an account known to belong to you) and social (eg. personality traits known to be associated with you). Gaining anonymity with Tor requires understanding these methods and how to counter them. See, for example, the Tails warning page.
If you were to use the TOR Browser Bundle, you would be using a browser that starts fresh every time that it runs with a new IP. It will also block plugins to prevent your real IP from being discovered through them. However, you could get banned for using TOR to have an account, because someone else may log in with the same IP address or they could just block TOR exit nodes to prevent abuse.
The internet works in a set of layers. Each one supposedly independent of one another. Up at the top, where the session layer sits, the cookies are meant to maintain the identity of the web user. Since IP addresses can shift and change, what people are really interested in is the identity associated with the cookie itself, which is why you see so much effort in creating and maintaining persistent cookies - these are the tokens that enable everything on the interactive web to work.
The use of the tor networks do not provide full anonymity. They provide good enough security but are not completely anonymous.
Yes, they can see your IP address and even more but that's not because of their cookies but rather other avenues (e.g, your headers sent from browser).
However the words "persistent or permanent" + cookies add a deeper cause of concern, because this means that even if you set your browser to delete cookies after each close or manually tell it to delete cookies, the cookie will regenerate itself again even without visiting the website again (I could give you a deep technically explanation for this but I won't - if you want to know more about this read about on Wikipedia).
No, using TOR will not add any benefit but rather call more attention to you and as Wikipedia states "In 2013, a top-secret NSA document was leaked by Edward Snowden 3 citing Evercookie as a method of tracking Tor users." If you log on your account via TOR they'll know it is you...
But don't give up on your privacy yet! There are some things you can do to delete these nasty cookies.
Unfortunately I'm too tired to write a tutorial for Firefox and Chrome now so I'll add links where you can see step by step what you can do and anyone feel free to edit my answer and add it if want.