What are the consequences when large-sample asymptotics does not hold? Lots of (frequentist) statistical inference is based on large-sample asymptotics. 
What are the specific consequences when a sample size is not big enough for the sampling distribution of an estimator to be approximated using the asymptotic sampling distribution?
The confidence interval will not have proper coverage, but will it be too wide or too narrow?
 A: Asymptotic confidence intervals on 'small' samples can have wider than nominal coverage, narrower than nominal coverage, or a mixture of both… sometimes a complicated mixture of both wider and narrower coverage.
(This is a broad question, so this answer is similarly broad.)
The particulars of wider vs narrower vs a mixture of both will depend on the specific distribution and the sample size in question, as well as by the specific form of CI. As one example, Figure 4 from Agresti & Coull (1998) below illustrates the empirical coverage probabilities based on simulations using two asymptotic CIs (left and center), and the exact CI (right) for a proportion (the $p$ of a Bernoulli distribution) at sample sizes 5 and 10.

As you can see, the Wald CI produces narrower than nominal coverage for virtually all values of $p$, the adjusted Wald CI appears to have less bias than the Wald CI when averaged across all values of $p$, but has a mixture, and the exact ('non-asymptotic') has strictly wider coverage. Also of note is the heterogeneous patterning of the biases across values of $p$.
So the answer is in effect it depends.
