Applications of lax 2-limits which are not pseudo 2-limits  One application of pseudo 2-limits (bilimits) in algebraic geometry is already found in the definition itself of stacks with values in a 2-category admitting bilimits (i.e. a discrete 1-cell-contravariant pseudo functor defined on a site). Of course there is the general notion of descent, etc. and all of its applications. There are also 2-colimit generalisations of aspects of Grothendieck's Galois theory (from SGA1) and aspects of SGA2.
The question is then whether there are "interesting" situations (not necessarily from algebraic geometry) where one must use lax 2-limits and cannot get away with just using pseudo 2-limits? (Or 2-colimits?) 
 A: First I should point out that (as you may know) in most of the 2-categorical literature, "bilimit" and "pseudo (2-)limit" are not used synonymously.  The former has a cone which commutes up to specified isomorphism, and has a universal property expressed by an equivalence of categories, while the latter also has a cone which commutes up to specified isomorphism, but has a universal property expressed by an isomorphism of categories.
The nLab uses "2-limit" to mean "bilimit," and so will I from now on.
Now, at least if one allows weighted 2-limits, then there is no situation in which one "must" use lax 2-limits and cannot get away with ordinary ones, since every weighted lax 2-limit can be identified with a non-lax weighted 2-limit which uses a modified weight.
However, lax 2-limits do occur naturally in some places, such as the Eilenberg-Moore object of a monad in a 2-category.  Note that weighted 2-limits which are neither unweighted ("conical") 2-limits or lax versions of conical 2-limits do also occur frequently, such as the comma categories mentioned in the comments.
