STV has been around since the 19th century, and in use in Australia and Ireland since 1918 and 1921 respectively. At no point have computers been involved.
Although it is simple to perform manually, the iterative nature of counting the votes, eliminating the least popular candidate and distributing their preferences can add extra time to arrive at a result in a tight race. In general, though, the margin between candidates and preference flows are predictable enough that it's rarely significant.
The algorithm to run an STV election is not that hard to implement. If the government is willing to release the ballot files, then there's no reason that you couldn't conduct your own count. But let's not pretend that we could do massive amounts of tallying without any kind of computer.
If anyone wants to try their hand at implementing the count for a multi-member electorate by PR-STV, the 'formal preferences' files for the Australian 2016 Federal Senate Elections are available here: http://results.aec.gov.au/20499/Website/SenateDownloadsMenu-... (scroll down to "Formal Preferences").
Single-member STV elections are considerably simpler to count.
There's no need to use computers to conduct an STV count. The single-winner variant (variously called "Instant Runoff Voting" "Majority-Preferential Voting", etc.) is especially easy to count by hand. Counting votes in a proportional election under STV is slightly complicated by the transferring of surplus votes, but it can still be done by hand.
Although it is simple to perform manually, the iterative nature of counting the votes, eliminating the least popular candidate and distributing their preferences can add extra time to arrive at a result in a tight race. In general, though, the margin between candidates and preference flows are predictable enough that it's rarely significant.