Test for diethylether or tertiary alcohol?
As mentioned in the comments, using Lucas' Reagent is the best option in this case because you're dealing with a tertiary alcohol vs an ether, but let's consider your proposal as well.

In principle, it should work because an ether will not react with ethanoic acid whereas an alcohol would. However, the formation of an ester simply by the addition of ethanoic acid to alcohol is not likely to complete (you generally need a dehydrating acid like $\ce{H2SO4}$ to drive the reaction). In other words, when you put in the bicarbonate you will most likely see effervescence.

If you're thinking of looking for effervescence, using thionyl chloride $\ce{SOCl2}$ might be an option since it will react with the alcohol to give a sulfur dioxide gas and HCl:

$$\ce{R-OH + SOCl2 -&gt; R-Cl + SO2(g) + HCl}$$

You have to make sure the solvent is dry though.

In the specific case of t-Butanol vs diethyl ether, though, you could simply heat the solution slightly to about 40 degrees Celsius. If it boils, then you've got diethyl either. If it doesn't, it's t-Butanol.