If you're changing just CPU clock speed, leaving the other components as is then amongst those that would continue to read tapes correctly are:
The Vic-20 and the Commodore 64. In the case of the Vic-20 tape input is connected to one of the control lines of a 6522 VIA. Wave length determination is achieved by loading a timer on that VIA and checking its value upon the interrupt triggered by the change in the control line. So if the VIA retains its original clock, tape loading will work as per usual. The same is true of the C64, but substitute a 6526 CIA for the VIA. You'd have to change your disk signalling routines though, because they're entirely dependent on the CPU toggling lines.
The BBC Micro and the Acorn Electron. In the former case a serial ULA feeds a Motorola 6850 ACIA. In the latter case a custom ULA does the whole thing. But in both cases the CPU does whatever it is normally doing and is informed by interrupt of tape events.
The MSX would likely not be affected for many pieces of software, depending on the degree of overclocking. It picks the speed it thinks data is on the tape based on measuring the initial tone, and supports a range from about 600 baud up to about 3600 baud. Commercial tapes are likely to be at around 1200 baud because it's one of the two fixed speeds built in for saving, and is more reliable than the alternative faster 2400 baud. So you could probably double the CPU clock rate and the BIOS would still cope — 1200 baud would look like 600 baud.
Machines that would definitely be broken that I'm aware of, in all cases because the CPU directly polls an input bit and counts time for itself:
- the ZX80, ZX81 and ZX Spectrum;
- the SAM Coupe;
- the Amstrad CPC*;
- the TRS-80 Model 1.
On the Oric tape input is wired to CB1 on a 6522 but I've never disassembled the ROM to figure out whether that is counted against a 6522 timer like the Commodores or something else. So that could go either way.
*) with the caveat that on the 464 and 4128, the tape drives are built in. So if you're modifying the machine, you could try just also overclocking the speed of the tape drive and hope for the best in terms of analogue filtering. Definitely worth a go in the scenario where you've managed to overclock the CPU somehow. And the same comment goes for the Amstrad Spectrums.