Can arbitrary cohomology classes $w_1,\dots,w_n$ from $H^{*}(B,\mathbb{Z}_2)$ be Stiefel-Whitney classes of some bundle over the given base $B$ or there are some necessary relation which can be obtained from topology of $B$?
In particular, I'm interseted in real bundles over $S^1\times S^1$ and $S^1\times S^1\times S^1$
On
This depends on what you consider as a "relation". There are no nontrivial polynomial relations between them, which hold universally, as one can see by calculating the $\mathbf{Z}/2 \mathbf{Z}$-cohomology algebra of $BO(n)$, which is a polynomial algebra in the Stiefel-Whitney classes.
An elementary reference for this calculation is the book of Milnor and Stasheff.
There are many relations that hold universally involving Steenrod operations; in fact the Stiefel-Whitney classes are generated under the action of the Steenrod operations by the classes $w_{2^i}$. This implies, in particular, that the first nonvanishing Stiefel-Whitney class must have index a power of $2$, so for example if $w_1 = w_2 = 0$ then it necessarily follows that $w_3 = 0$.