The knob is too rusted to unscrew and replace? Can someone still place a lock on the door?

There are numerous products you can use to remove the rust--or to make the screws easier to turn. Liquid Wrench, WD40, naval jelly to paint on or spray on--go to your hardware store and look. Once the knob is removed, you can then replace it. If products that you apply don't work, use a drill and remove the screws by drilling them out. 

A deadbolt lock can be installed above the knob. The door does not have to be replaced.

There are always alternatives. Without any kind of picture it is ******* impossible to say.

"not possible" is false.  this can be done if you're willing to destroy the knob and mechanism -- after which a replacement can be fitted.

The knob can always be busted up and off one way or another.

You can put a lock that isn’t attached to the doorknob. You didn’t post a pic or describe the type of door or lock you’re talking about. If it’s a bedroom or bathroom, you can use a hook & eye, or a sliding deadbolt. I’m sure there’s a way to get the old knob off though. It’s hard to help you with just that little description you gave. 

If you are a DIY kind of guy, the knob itself can be replaced by simply drilling out the heads of the screws that hold the knob together.  Once apart, the entire thing can be replaced easily.   For added security, until you get the knob replaced, you can add a deadbolt above the knob. 

Get your drill and a good drill bit and drill the heads of the screws off and the lock will then come apart.

Your not trying hard enough.  There's always a way to remove something. If there stripped you can get screw extractors.   Or drill the screws out. Would be easier to suggest options if we could see a picture of what your issue is.

I bet I could get it off with an angle grinder

No knob is too rusty or anything else to get off.  If you can't there are plenty of other folks who can.  Ask someone you know who works on cars, or is a carpenter or a janitor.

It depends what sort of knob, but in general, if the knob and screws are very rusty, just get a hammer and chisel or screwdriver, hammer it under the plate where the screws are, and pop the screws out. Then the whole thing will fall apart and come out. 
You do not need to replace the door. 

USA today                

See if  the knob can be beaten off by using a hammer and screwdriver up against it to get it off.

cut it off with a hack saw or a grinder 

It is literally IMPOSSIBLE for the knob to be too rusted to replace. In fact, the more rust, the EASIER it is to remove it. That said, adding a deadbolt above the knob is NORMAL. With few exceptions, you can drill a hole for the deadbolt in any door worth locking.

Nothing is EVER TOO RUSTED.  A carpenter or handyman can get the knob off and replace with another knob.  Because that is the cheapest route to go....even if you have to pay the carpenter money.  THEY HAVE THE TOOLS & KNOWLEDGE.  You don't.

try bit bit of wd40 and some muscle , the should help .or maybe its a false knob , does it turn ..