Suppose a true model is $Y_i=\beta X_i +e_i$, where $e$ is the random error. Suppose instead we fit the model (using least squares) as $Y_i=\alpha_0+\alpha_1 X_i +v_i$, where $v$ is the random error.
What are the consequences of this specification error on the model?
This is not model mis-specification. Your true model is still within the model you're fitting, simply with intercept equal to zero. So any results that hold for linear model with an intercept continues to hold here.
You may wish to compute the statistical efficiency of both estimators, by working out their Fisher's informtion. My guess is that the inclusion of an extra intercept term does not affect the efficiency.