Basic RF theory tells us that it is the noise figure of the RF amplifier that determines the overall noise floor.
The following is a basic example.
In this example the tuner/receiver has a noise figure of 3 dB which is a fairly good number (the lower the better).
The RF amplifier in the amplified antenna might have a noise figure of 5 dB (not an unreasonable number) and a maximum gain of 15 dB.
At maximum gain the overall noise figure will be about 5 dB. If the gain of the antenna is turned down to for example 5 dB then the noise figure is about 5.4 dB. Note that the noise figure has increased and a lower number is better.
Here is the math if you want to check the numbers.
As long as the noise figure of the RF amplifier in the antenna is greater than the noise figure of the tuner/receiver, it will degrade the overall noise figure. Unfortunately the noise figure of amplified FM broadcast band antennas is not usually published.
There is no substitute for a full sized antenna for the FM broadcast band. If using a physically small amplified antenna improves one's reception, that means that the electronics (the FM amplifier) in the 30 dollar antenna is better than the electronics in the tuner/receiver. See the math above.
If the noise figure of the RF amplifier in the antenna is less than the noise figure of the tuner/receiver (a good thing), reducing the gain of the RF amplifier in the antenna will still result in an increase in the overall noise figure. For example.
Noise figure of antenna amplifier 3 dB. Noise figure of tuner/receiver 5 dB. These are realistic numbers.
If the gain of the antenna amplifier is 15 dB, the overall noise figure is about 3.2 dB and an improvement over the basic noise figure of the tuner/receiver alone (5 dB). Remember the lower the number the better. This means that there may be an improvement in the reception of a weaker station.
If the gain of the amplifier in the antenna is reduced to 5 dB the overall noise figure raises to about 4.3 dB, still slightly better than the tuner/receiver alone, but it is still an increase and remember the lower the number the better.
Another variable that may come into play is the quieting slope of the FM tuner/receiver, but that is another can of worms.
The bottom line is that is not easy to accurately predict the result of an amplified antenna with unknown specifications when used with tuners/receivers with unknown performance. These antennas are cheap enough to give one a try, although it may or may not be an improvement. They will not usually cause any damage.