My school has recently asked us to submit our MAC address to the school along with our designated name to be used to connect to the Wi-Fi. Previously this wasn't needed. 

I would like to ask about what kind of information that they can collect from this? Would they be able to track our browsing history or more? What if I use Tor Browser? Would it have any effect?

If they can track me, what measures can I take to prevent them from invading my privacy? 

I think you should ask why they want to use the MAC address, not necessarily for privacy reasons; "why do you need the MAC Address?" I think it's a reasonable question to ask them.

Firstly, they will have MAC addresses of all the individuals who connect to the WiFi. Any device connecting to the WiFi will reveal their MAC address, based on the ARP protocol. 

They may think locking down WiFi to known MAC addresses is a good security measure. It's not really because I can obtain your MAC address if both of us are in the same Starbucks and on the same WiFi. I can then spoof your MAC address quite easily. So from a security measure this is not great.

They may want to track your activity. They can do this already without asking for your MAC, just giving them the MAC address allows them to map it to a individual easier. They can get a history of MAC & IP address from logs and  their NAT can keep a history of IP Address & Ports and map back to the MAC address.

If you use Tor, they will be able to say you used Tor, but not the content.

So, I would ask why do you want my MAC address, giving out the MAC address is not going to really affect you. Unless of course on your home WiFi or something else you are using MAC address as a method to identify yourself; as MAC address can be easily spoofed. 
