Turing machine - infinite tape - does that thing exist?
The tape is not infinite. It is unbounded. 
There is an important difference.
We don't need infinite memory.
We only need to be able to add more memory over time as needed.
This is actually pretty close to what people do in practice.
When a program needs more memory it asks the operating system and
receives more memory.
If we see that a computer is short on memory we increase its memory.
It is becoming even easier with cloud computing.
So the assumption is not that the machine has infinite memory.
The assumption is that we can increase the memory as needed.

If your question is about the limit where 
the needed memory reaches the limit of information 
that can be stored in the universe 
(assuming that it is finite and bounded by some fixed number)
then yes, we will not be able to add any more memory at that point.
In fact, if you assume that 
the amount of information is bounded by a fixed number
then the world is a finite state machine.
However that is not the usual case.

However if you try to design programs as finite state machines 
you are going to have a very difficult time
because the number of states is huge.
The number of states for a Turing machine with $m$ bits of memory 
would be something like $2^m$.
Turing machines (and other similar models)
simplify things.
This is a rather common thing in sciences.
It is important to know the limits of a model but
the fact that a model is not a completely accurate representation of the world
doesn't imply it is useless.