In an OLS regression, will excluding all data for a non-reference category of a dummy variable impact the other dummy level categories? Say I have an OLS regression with a dummy variable level A, B, C and D, where A is the reference category. Will the estimated coefficient value and/or statistical significance of B or C change or be impacted if I remove from the input data set all of the data mapped to D and re-run the regression (which will no longer contain dummy variable level D)? 
On the values changing or not, I would think not because the coefficients of B and C simply reflect the difference between the mean of B and C, respectively, and the mean of A (the reference).
Is my understanding correct?
 A: I interpret what you are doing as removing all data associated with the label 'D' from the dataset before fitting the model. This is distinct from relabelling 'D' to 'A' or some equivalent step, which is Noah's assumption. 

It depends on whether you have any other variables in the regression. If you do not, then your interpretation is correct. Dropping D will not affect the intercept (A), or the coefficients for B or C. 
But imagine you also have an additional variable that you are using as a predictor. If the distribution of this continuous variable is non-random with respect to D vs. A, B, or C (i.e. they are associated in some way), then dropping D may also change the coefficient for the additional variable. This change may also affect the coefficient estimates and associated p-values for A, B, or C if they are associated with the additional variable. 
A: Edit: OP clarified their question, so this answer does not apply. This answer responds to the question of simply removing the dummy variable for category D from a regression that contained the full dataset.

Dropping categorical D is equivalent to relabeling all D's as A's. This will absolutely affect your intercept and coefficients for the non-reference categories. The intercept will now be the mean of those in categories A and D combined, and the coefficients on the non-reference categories will be the difference in means between the corresponding category and the combined A and D group. The only reason it would make sense to do this would be if you wanted to assume that the means for categories A and D were equal. If they were equal in the population, you would have some increased precision in your estimates. If not, you will have biased estimates for the true coefficients. 
