Significant digits are in the curriculum for both my Chemistry and my Physics class, and all my Calculus students have taken at least Chemistry. It also pops up in the school's Precalculus class (which I do not now teach). Although significant digits are not in the Calculus curriculum I get many questions about them.
Should I continue my present practice of ignoring sig digs in my Calculus class (and stating the desired precision of approximate answers in quizzes and tests) or should I include them in some way?
Note that I am not asking if sig digs should be used in science classes: that is a given for now.
No. If you are going to have questions that have decimal/numerical answers, it makes sense to decide on some particular place value for students to round to for the sake of uniformity, but basing this off of significant digits reflects a misunderstanding about what significant digits are used for.
Sciences use significant digits because problems are posed in terms of experimental givens, where the amount of listed digits loosely implies confidence in measurement, for example a 1 kg vs 1.00 kg mass. Significant digit rules make sure that the answer implies the same level of measurement accuracy as the given quantities. This is not normally considered in math classes because the given quantities are assumed to be mathematically exact/ideal, since they are just "invented" for the purpose of the exercise and not measured.
Most of the methods of calculus are at first directed towards describing the exact formulas for differentiation and integration. In these cases I don't really see the point of sigfigs. If you were interested in such things, maybe talk about error in a slightly more sophisticated way by describing how the derivative of a function can be used to propagate error in input. The is, we have data $x$ with error $\pm \delta$ then we can approximate error in a function $f$ by $f(x) \pm |f'(x)|\delta$ of course, it should be emphasized that this is not the only source of error (error may be in the model/function not the data) nor is it perfect (for example it is absurd to claim error at a max is $0$.