Let \(X\) be a complete \(\mathbb Q\)-factorial toric variety with tangent sheaf \(\mathcal T_X\). The main result of the paper gives a combinatorical description of \(H^2(X,\mathcal T_X)\) and the cup product map \[H^1(X,\mathcal T_X)\times H^1(X,\mathcal T_X)\longrightarrow H^2(X,\mathcal T_X).\] As an application an example of a smooth toric threefold is given for which the cup product map does not vanish, i.e. smooth complete toric varieties may have obstructed deformations.