How to grep a log file from specific date to end of file?

Provided there are fewer than 999,999 lines:

grep -A 999999 '19/Jan/2016:22:' access_log

But this would be a better solution as it doesn't restrict the number of lines after the match:

sed -n '/19\/Jan\/2016:22:/,$p' access_log

use sed instead:

sed -n '\#19/Jan/2016:22:#,${/<regexp>/p}' access.log

where <regexp> is a sub-expression you would like to find in lines after 19/Jan/2016:22: