Suppose I have a vector bundle $V$ over a smooth projective variety $X$ with the following exotic property: any global section of $V$ does not vanish anywhere.
The general question what can we say about this vector bundle? For example, it looks like it is not necessary generated by global sections: it could be not enough global sections. But can we claim that it has a free direct summand of rank $h^0(X, V)$?
But mostly I'm interested in tangent bundle $V=T_X$. Could someone give me an example of $X$ s.t. $T_X$ has this property, but is not trivial as a vector bundle? Is this possible?
Here is an example of the type you require:
If $Y$ is a connected smooth projective curve of genus $g\geq2$ its tangent bundle $T_Y$ has degree $2-2g$ and has thus $0\in \Gamma(Y,T_Y)$ as only regular section.
On the other hand an elliptic curve $E$ has trivial tangent bundle $T_E=\mathcal O_E$.
Thus the product $X:=Y\times E$ has as tangent bundle: $$T_X=p_Y^*(T_Y)\oplus p_E^*\mathcal O_E=p_Y^*(T_Y)\oplus\mathcal O_X$$ and $\Gamma(X,T_X)=\Gamma(X,p_Y^*(T_Y))\oplus \Gamma(X,\mathcal O_X)=0\oplus k$ .
So each non-zero section of $T_X$ vanishes nowhere, just as you required.