Will installing a firewall affect a TeamSpeak3 server in any way?

As mentioned if you’re going to be following the guide you linked then you’re setting up the firewall to block all incoming traffic, except for on specified ports, which will affect teamspeak.

However (assuming you’re using the TS3 default ports) you should be able to run the below to open the necessary ports:

sudo ufw allow 9987/udp
sudo ufw allow 30033/tcp
sudo ufw allow 10011/tcp

The point of a firewall is to close ports, so yes it will effect, cause by default all ports are open, with the firrwall just close all ports except the ones you need,

For example if your running a web server then you would open port 80, teamspeak3 requires 3 different ports just open those ports the way ts3 needs and it will run fine.