I anexed the report from OpenHardwareMonitor. Under sensors, going to the AMD Radeon HD 8670D + R7 240 Dual Graphics tab, you can see that my GPU fan is doing 749652 RPM while around 28ºC. Is this rotation value too high or is it normal? The fan is so loud right from startup and it doesn't stop being loud ever.
According to MSI Afterburner, the fan is currently at 30% and when I change the value to 100%, it definitely gets louder, so it doesn't seem to be running at full speed. Also I opened the case and started the computer and when I put my finger on the case around the fan, the noise is less intense. It feels like the whole graphics thing is vibrating. Any thoughts?
Which are you referring too? the APU CPU Cooler fan or the R7-240 GPU Card fan?
Generally, if the GPU or CPU Cooler fan has become noisier than the way it was in the past, seems to be indicating that possibly the bearings are wearing out.
If the R7-240 GPU card is fairly old than it is possibly the fan is beginning to wear out and become noisier.
Anyways, whether it is the CPU Cooler fan or the GPU Card fan, if the fans starts to get noisier and louder then when you first had it then it probably is worn out. Try if possible, to put a tiny bit of oil on the fan's bearings if you have access and see if the noise lessens.
Otherwise either try to replace the noisy fan or wait til it completely fails which is not advisable.
Forgot to mention, check to be sure there are no cables or obstruction touching the fan blades. Also clean the Fan as best as possible in case there is dirt/dust accumulating inside the fan's casing.
It's definitely the GPU Card fan. Which oil should I use?
First you need to see if you are able to apply the oil to the fan.
Click on this YouTube search page that has several methods of lubricating or repairing noisy or loud or slow GPU Fans: apply oil to gpu fan - YouTube
From above link some examples on fixing failing GPU Fans:
just a note of caution, if your GPU is still under warranty open a RMA with the GPU card manufacturer first. They might be able to send you a replacement fan or Cowl under Warranty or sell you one if they still have it in stock.
Otherwise Good luck and I hope you are able to fix it to last a few months longer.