You can test without a resistor for the moment, and set the volume lower if you hear distortion.
Power into the speaker will depend on the sound you play, and the volume you play it. The speaker you’ve chosen has a maximum printed on the back, and is said to be from stock that is labelled either 0.25W or 0.5W.
Should you deliver more power to the speaker than it can handle, there are two results;
- the moving cone will hit the limits of movement and distort the sound, and;
- the temperature of the coil may rise and cause damage to the speaker.
When you hear distortion, you’re probably sending more power to the speaker than it can cope with. However, the converse is not true; when you don’t hear distortion, that doesn’t mean you can send more power.
Not a criticism, but the player module doesn’t have very clear specifications on this. The best it has is a 3W maximum implied by the words drive speaker less than 3W. So if you used a speaker of more than 3W you would not risk damage to the speaker. On the other hand, the module looks too small to be able to deliver that amount of power to the speaker, and would need at least 600 mA supply at 5V, which can be a challenge from an Arduino Uno; you can measure the supply current with a multimeter and use P=VI to estimate the speaker power.
Not a criticism, but the speaker doesn’t have very clear specifications either; it doesn’t say what sort of temperature or airflow the limit is set for. It doesn’t say what frequency of sound was tested. In my experience, this is usually 21°C, still air, and 1 kHz tone. Lower frequency tones cause more heating in these speakers.
You can limit the sound level in the settings. This will reduce the power, and the heating.
You can add a resistor if you like, but it will have to be at least a 2.5W rating, otherwise it will become the first thing to heat up and fail.
Here’s what I would do. Use Audacity to make a few test tone files, at 500 Hz and 1 KHz, four seconds long, set the module to full volume, give it 5V via a multimeter set for current, and see what sort of current you get when playing the test tones, starting from low volume and slowly increasing. Should you see no more than 50 mA, then it will be fine to use a 0.25W speaker.
Hope that helps!