What are reasons to believe that e is not a period?

Periods arise from the comparison between Betti and de Rham cohomology for an algebraic variety. The Period Conjecture, due to Grothendieck, is a transcendence conjecture for periods which says that every algebraic relation between periods arises from geometry (in a certain precise sense).

More generally, there is a wider class of complex numbers called exponential periods arising from the comparison of rapid decay cohomology and de Rham cohomology. The number $e$ is an example of an exponential period. There is an analogue of the Period Conjecture in the setting of exponential periods, and the truth of this conjecture would imply that $e$ is transcendental over the ring of ordinary periods (see Proposition 10.1.5 of the paper Exponential Motives by Javier Fresán and Peter Jossen). So the exponential Period Conjecture provides a heuristic coming from algebraic geometry that $e$ is not a period.

To my understanding, the reason is simple: in the almost 300 years since $e$ was discovered, no representation of it as a period has been found. I think this is quite a strong evidence.

Remark. To those who think that periods were introduced by Kontsevich and Zagier, I recommend the paper of Euler, On highly transcendental quantities which cannot be expressed by integral formulas English translation. (Strange that he does not mention his own $e$ as a candidate. Perhaps he was still looking for an integral that equals $e$ when he wrote this paper.)

One very weak heuristic comes from the continued fraction expansion $$e=[2;1,2,1,1,4,1,1,6,1,1,\dots],$$ which exhibits a very simple pattern. If you exclude rational numbers and quadratic irrationals, which have finite and periodic continued fractions respectively, the rest are expected to have "generic" continued fractions. In other words you expect their continued fractions to exhibit properties of "almost all" real numbers. One such property is the convergence of the partial geometric means to Khinchin's constant. For example this property seems to be satisfied numerically by $\pi$.

I guess this is in the same vein as the heuristics that non-rational algebraic numbers (or even periods) are normal in every base.