The mummy's coffin that is shipped from Cairo is obviously different from the one that arrives at Professor Bitts' home. The writing on the coffin that arrives at Professor Bitts' home is neater, the "Paid" stamps are in different locations, and the twine around the coffin is in a different location.
The speed at which the Saint writes notes bears no relationship to the measured writing on the actual pieces of paper. This is especially noticeable when he scribbles on the top of The Philadelphia Post newspaper with a pencil, whilst the newspaper is unsupported. The next shot shows the paper is perfectly flat on a stable background and the message he writes is neat and even. Although the Saint ends the message, the following shot shows him still finishing writing as he tears off the strip of paper to hand to the paperboy.
When a bullet is shot through the side of the speedboat carrying Templar, the depth of the hole from the inside angle closeups is not as low as it would have to be near the waterline to take on water as obvious from the medium shots of the outside of the boat.
The original logo at the beginning of the film was the commonly seen post-1936 "RKO Radio Pictures" style, but it apparently was missing from the material provided for the TCM version, and an inappropriate pre-1936 "Radio Pictures" style logo was lifted from a much earlier film and used in its place.
When the Saint is in the speedboat, there is water spraying alongside the boat, implying the boat is moving at great speed, but the water just beyond shows it isn't moving at all - it's just being agitated off camera.