It depends on what kind of animation you mean by "Flash-like" animations.
If you mean "a lot of spinning doodads and scrolling marquees that do nothing," then your client is almost objectively wrong (insofar as any answer in a creative field can be "wrong"), for the reasons mentioned in the answer contrasting the images of a gaudy bus and a Porsche.
However, if your client is referring instead to jQuery/HTML5-type animations that indicate events on a page (e.g., when a user proceeds to the next step in a process, the screen visibly scrolls to the next step), then consider that your client's perception could be right on the money, at least for some users (e.g., mobile users are probably more used to interaction animations, and might even expect them). Your job in this case would be to explore which animations are important (i.e., which animations convey information to the user) versus which animations are superfluous.
Let's say that your client wants the first type of animation, the gaudy and useless ones. Persuade the client to revise his ideas by rephrasing your objections. Instead of "this animation thing is a bad idea because XYZ," instead say something like "I've reconsidered your animation ideas, and you're actually right that the site would be boring if it isn't animated - we can incorporate animations in such and such a way." Then reinterpret the spec so that the animations are the user-friendly usability-enhancing type of animations (sliding transitions and the like), instead of useless idle animations. If necessary, stress or exaggerate the limited screen space and system resources of some smartphones and tablets to cut down as far as possible on the useless animations. When you have finally succeeded, be prepared to laugh off the client's inevitable "I told you so" when you do manage to make a sensible site out of their overly flashy spec. The more the client feels like they got what they wanted, the better, regardless of the consequences for your patience and/or pride.