When Turing Machine behaves like Finite state automaton I read following:


  Turing Machine with finite (fixed sized) tape is essentially Finite state automaton.


Is this fact correct? My doubt is Turing Machine can go infinite loop even on finite tape if it keeps moving from left to right and finite state machine does never go in such infinite loops. So, do we also need restrictions like "only right head movements are allowed" along with finite tape to turn Turing Machine to finite state automaton?
Yes, this is (kind of) correct, in the following sense:

The set of languages that can be recognized by Turing Machines with a fixed tape is equivalent to the set of regular languages (i.e. the set of languages recognizable by finite automata).

The reason behind this is that such Turing Machines have a finite number of configurations, which is independent from the input. 

Note that crucially, for this to hold we need one tape for the input which is read-only, and a fixed-space working tape.

A bit more precisely, for such a TM, we can construct a 2-way DFA, by having a state for every possible configuration of the TM, as well as a 2-way read only input tape. The latter model is equivalent to DFAs, by a clever construction using "crossing sequences".

An important observation, however, is that these machines can be far more succinct than their equivalent DFAs (or even NFAs).