1. It suggests that you can shift one curve without the other moving as well. In other words, it assumes that excess demands in the goods market are independent from excess demands in the money market.
2. The IS curve — which involves investment demand — uses the real rate of interest, r. The LM curve — which involves money demand — uses the nominal rate of interest, i. These two curves are then put on the same graph. I have been told many times this can be done without contradiction; at best this is true only in the shortest of runs, when prices are not changing.
3. Everything in the model is flows, but stocks matter too. No person in the model considers his or her intertemporal budget position. You don’t have to believe in Barro’s Ricardian debt-equivalence theorem to be worried by this.
4. Coordination problems — which should be at the center of macro — are obscured by the aggregate apparatus.
5. The IS curve, drawn from investment demand, uses the long-term rate of interest. The LM curve, drawn from money demand, uses short-term rates of interest. Yet the relative movements of short and long rates remain a significant puzzle and do not follow the predicted relationship.
6. You see the curves — which remind you of supply and demand curves — and you wish to start manipulating them in the same manner. See #1.
My (significant) concession: The model does fairly well predicting many economic phenomena in the short run. But you could do better reading Arthur Marget on sophisticated versions of the quantity theory of money. I hope to explore this point in more detail soon.