Common false beliefs in theoretical computer science
This is one is common to computational geometry, but endemic elsewhere: Algorithms for the real RAM can be transferred to the integer RAM (for integer restrictions of the problem) with no loss of efficiency.  A canonical example is the claim “Gaussian elimination runs in $O(n^3)$ time.”  In fact, careless elimination orders can produce integers with exponentially many bits.

Even worse, but still unfortunately common: Algorithms for the real RAM with floor function can be transferred to the integer RAM with no loss of efficiency.  In fact, a real-RAM+floor can solve any problem in PSPACE or in #P in a polynomial number of steps.