A person, who is not a Sheriff, has been trying to serve me with a Child Support Subpeona. If they are unable to subpeona me for court and I do not make the court date, will I than have a warrant?

What State? You may or may not have to be personally served. After a certain number of tries it can be nail (to your front door) and mail (to your address) or substitute service or anything else. A Sheriff does NOT have to serve a Subpoena.

Again - where?

I question the if you do not make the Court date statement - in theory you don't know when the Court date is so, of course, you won't be there.

And you can be served in the middle of the night, at work, on the street, at a relative's house.

I'm a process server. I hope "this" isn't one of mine!

Normally they will use some other method of service, according to your location rules.
Then they will just have the court case without you, They will take anything the other side says for fact and you will just lose.

So if you want to protect your rights in a court of law, you need to get served, Since it will be the worst possible outcome if you are not.