i have a deep sink too, and all plumbing is connected w/o issues. So to answer your ?, it should not.
What can happen is that if you store a waste basket underneath, than you should check if the basket will fit.

Examine the vertical clearance from the bottom of your existing sink to the input of your P-Trap. If you have a vertical length equal too or greater than the difference in depth between the old and new sink you will have no problem, you just shorten the sink tailpiece a bit, however if your drains have no available vertical clearance it may require physically lowering the waste arm coming out of the wall. (This would be an extremely rare circumstance).

Thank you for easing my mind