What is the oldest open problem in TCS?
What is the computational complexity of integer multiplication?  Arguably, this question dates back at least to the the 'duplation and mediation' algorithm for integer multiplication described in the Rhind Mathematical Papyrus, which was written circa 1650 BC, but claims to be a copy of a significantly older document.

Admittedly, the Rhind papyrus did not explicitly consider the complexity of is algorithm.  So maybe a better answer is What is the complexity of solving systems of linear equations?  Research into efficient algorithms for solving linear systems dates back at least to Liu Hui's commentary, published in 263 AD, on The Nine Chapters on the Mathematical Art.  Specifically, Liu Hui  compares two variants of what is now recognized as Gaussian elimination, and counts the number of arithmetic operations used by each, with the explicit goal of finding the more efficient method.

Both of these questions are still targets of active research.