L1 penalized regression introduces a bias on your regression model but decreases the variance. When this bias is introduced, is it possible that the coefficient of $B$ changes sign? This would greatly affect the interpretation, where a regular regression might tell you an increase in $x$ increases $y$ by $B$, but maybe using lasso an increase in $x$ decreases $y$ by $B$
Yes, the sign of a coefficient can change. For a reference, in the original paper of Tibshirani (1995) it says that the signs of the LASSO estimate and the least squares solution may be different (p270 last paragraph).