I know what you mean, that no one dies but new people are born and the population excedes the resources needed to sustain it. But there are a few factors that prevent or slow down this result.
Firstly, Aman is quite a resourceful land.
Secondly, Aman is large. I believe Elves only inhabited parts of the eastern shore. They went everywhere, I would assume, but their centre seems to be just in the middle. If
this map is to be trusted, even Formenos, that is said to be farther north than most Elven settlements, is still in the centre area. The main settlements - Tol Eressea, Alqualonde, Tirion, Taniquetil, etc - are all grouped together, leaving vast lands to the north and the south.
Also, Elves have the ability to make the most out of the least when it comes to food. Take lembas as an example. Perhaps they did not all feed on lembas, but they would not go hungry.
Then the Elves (except for Feanor*) don't tend to have many children. When you have millenia to live you don't rush with marriage and children.
Moreover, there is the whole thing of most of the Noldor leaving to ME with Feanor and Fingolfin (and most Teleri slain in the process). This would clear some breathing space for a couple ages (and the Halls of Mandos are never too crowded). It's an issue when they start returning, but they don't all come back at once, but gradually. And quite a few returned via Mandos.
Last but not least, Elves do die, even in Valinor. And some don't come back *coughmirielcough*
*Well I guess his half-brothers could go there as well... And according to some Finwe had three sons too many...