We certainly have the capability to build islands. China is doing it right now, although to a much lesser extent than islands the size of Massachusetts. It would be a massive project that would cost a great deal. But if we deemed it necessary we certainly could do it.
Whether or not we should is a separate issue.
1. How long, at what cost, with how many diggers would it take to do this ambitious project to any real affect? Expect very high numbers for all three.
2. The mountains stop the weather destroying us all.
3. Many people already live in the hills and mountains. All this does is destroy their homes. It'd be years before the islands were constructed, and more before living areas would be prepared there- it's just more left homeless.
4. With the money used trying to destroy mountains for land, we could set up huge underground areas or areas out at sea in half the time.
Eh, my city has gone waaay past rock fill for reclamation. These days, it's mostly marine fill plus some public fill thrown in (since there's little use for the waste construction materials elsewhere).
Rock fill is incredibly expensive, and given the scale of your project, it's reasonable to say that your project will not be cheap by any measure. There will be a... Giant stimulus on the economy if you do start this, but the government doesn't have unlimited reserves, and it's not like we really need an inflationary gap.
Rock fill causes a myriad of horrible environmental consequences. At the surface, there's the obvious visual pollution caused by deforestation, noise and air pollution arising from machinery that removes the rocks and regolith. Clear-cutting also leads to soil erosion and thus silting in nearby bodies of water, which in turn increases the risk of flooding. Habitats are destroyed as well, and if there are indigenous (or non-indigenous, for that matter) people living in the area, good luck with the relocation and compensation.
There are much better approaches to dealing with this problem, such as developing cities in the compact urban form...
If the purpose is to create additional living space for an over-populated planet, all this does is to postpone the inevitable, and encourage more over-population in the process.
Tearing down mountains and building islands, get ready for some planetary climate changes that might well kill more people (and other species) than the new islands saved.
Also, exactly whose mountains do you propose be stolen from them to do this endeavor? I can think of some places where this would result in all-out war. Which would reduce the population enough to no longer need the islands.