If you have GNU tail available on your system, you can do the following:
tail -n +1000001 huge-file.log
It's the +
character that does what you want. To quote from the man page:
If the first character of K (the number of bytes or lines) is a `+', print beginning with the Kth item from the start of each file.
Thus, as noted in the comment, putting +1000001 starts printing with the first item after the first 1,000,000 lines.