Well, in principle. It wouldn't sink into the sun, unless you use a interstellar lithobraking for all your souvenirs to deliberately do it, but you could perfectly well turn Earth into a black hole that way.

Of course it's not very practical, and also it'd turn into a star much earlier in the process, at which point it arguably isn't a planet anymore anyway.

Are there any nearer-term concerns? Imagine we have large scale asteroid mining within the next century. Is there a point where the extra-mass matters on any reasonable timescale (hundreds of years) similar to the impact of industrialization with Co2 and climate change? Other odd side effects from landing the material?

No, no near term concerns at all. It’s a very silly sci-fi scenario. You could smash the mass of Mount Everest into Earth a hundred times a day for a thousand years and it wouldn’t even increase the planet’s mass by 1%.