Do we need to read the input completely in Turing machine?
There is no such requirement. A Turing machine which decides a language can do whatever it wants as long as it always halts; the language it decides is the set of words which it accepts. 

There are many languages which don't require reading the entire input:


The empty language.
The complete language.
The language of all words starting with 0.
The language of words of length 2.


Here is a more non-trivial example:


Given a language $L$, define the language
$$ X(L) = \{ 0^\ell 1 w x : |w|=\ell \text{ and } w \in L \}. $$
The language $X(L)$ can be decided by a Turing machine which doesn't always read the entire input. Moreover, generally speaking $L$ and $X(L)$ have roughly the same complexity (though $X(L)$ requires the ability to count).