Recently, a blog was posted that explains that the President of the Senate (Pence) has the sole authority to decide which certificates of election to open and count. It points out that there was a dispute in which Georgia's certificate of election was clearly defective, but Jefferson (President of the Senate at the time) chose to count it anyway.

The top comment on the post said "This is actually not true. It was contested & the house had voted 26 times (always in a tie) before rejecting him." This partially true. It is an indisputable fact that Jefferson chose to open and count Georgia's votes despite the defective certificate. If he had not done this, nobody would have had a majority of Electoral votes, and there would be been a contingent election BETWEEN ALL FIVE CANDIDATES WHO RECEIEVED ELECTORAL VOTES. The House of Representatives was Federalist, so John Adams would have won.

However, because he did count the votes, he and Burr had a majority, BUT they were tied. (At the time, each Elector had two votes, and the first place became President and second place became Vice President.) So, there was a contingent election BETWEEN THE TWO OF THEM to decide which would be President and which would be VP. It was a tie for thirty-five times before Hamilton convinced some Federalists to vote for Jefferson in order to prevent Burr from becoming President.

TL;DR: The article is correct that the President of the Senate chooses which votes to count. Jefferson chose to open and count Georgia's certificate, but that left a tie between him and Burr, leading to a contingent election. Mike Pence can choose whether or not to open and count any State's Electoral votes.