When can the impulse response become zero?
The notion of sparsity entails that an object, living for instance in an $n$-dimensional space, can be described (in the suitable basis/frame) by a number $k$ of meaningful components (each above a threshold, or whose combination is close enough to the signal)  "much smaller" than $n$.

When talking about filter identification (adaptive or not), the filter is, at the beginning, unknown. To avoid looking inside an infinity of possibilities, one first restricts  the space of solutions to the subspace of length-$n$ or order-$n$ filters. Either for true physical reasons, out of wishful thinking or heuristic motivations, or because  the optimisation problem is not directly tractable (non-smooth, non-convex), one may call for Ockham's razor, or the law of parsimony, one may want to restrict the solution (not the real object) to be $k$-sparse.

Your example could  be sparse per se, if initially the space of solutions could have up to 100 taps. Additionally, it could perhaps be approximated (in practice) by a sparser filter:
$$y[n] = x[n] + 0.6*x[n-3]$$
discarding the two smallest coefficients.

With this introduction, let us go back to your questions:


I would disagree with "FIR filter = Moving Average and IIR = Autoregressive model": an IIR can be AR/MA, some FIR can be rewritten in an AR fashion. The rest of the question is unclear to me: if the model is linear, the impulse response would be some sparse approximation of the actual coefficients.
The actual coefficients are not becoming zero. But, either they form a  filter short enough to be qualified as sparse, or their estimate has very few non-zero taps
It may happen in some echo cancellation problems: either in a room or on delay lines, one expect that the source bounces after some travelling back and forth. If at each bouncing the reflection is scarce, and there is some attenuation, the  "global" echo filter could look like  (two bounces)
$$y[n] = x[n] - 0.25*x[n-99]+ 0.5*x[n-100]+ 0.25*x[n-101]  + 0.125*x[n-199]+ 0.25*x[n-200]+ 0.125*x[n-201]$$
hence a sparse filter, even with order 201. Outside speech processing, we used such priors in seismic processing of remove multiples, approximate models of bouncing waves, assuming that the filters were sparse at boundaries between rock layers, see A primal-dual proximal algorithm for sparse template-based adaptive filtering: Application to seismic multiple removal, 2014, IEEE Transactions on Signal Processing:



  Unveiling meaningful geophysical information from seismic data
  requires to deal with both random and structured “noises”. As their
  amplitude may be greater than signals of interest (primaries),
  additional prior information is especially important in performing
  efficient signal separation. We address here the problem of multiple
  reflections, caused by wave-field bouncing between layers. Since only
  approximate models of these phenomena are available, we propose a
  flexible framework for time-varying adaptive filtering of seismic
  signals, using sparse representations, based on inaccurate templates.
  We recast the joint estimation of adaptive filters and primaries in a
  new convex variational formulation. This approach allows us to
  incorporate plausible knowledge about noise statistics, data sparsity
  and slow filter variation in parsimony-promoting wavelet frames. The
  designed primal-dual algorithm solves a constrained minimization
  problem that alleviates standard regularization issues in finding
  hyperparameters. The approach demonstrates significantly good
  performance in low signal-to-noise ratio conditions, both for
  simulated and real field seismic data.


Note that parsimony, usually resorting to an $\ell_0$ count measure, is not easy to optimize, as non-differentiable, non-convex, etc.