How to check whether a component can work in strong magnetic fields?
I want to design my PCB to operate nicely even when we place it next to a neodymium magnet. How to check whether my component can work in such condition without shielding?


You can expect potential problems if a device contains a moving conductor, "magnetic material" or is designed as a magnetic or electric or electromagnetic field sensitive or field sensing device.

Magnetic field decreases with the inverse cube of the distance from the centre of the North-South dipole so it gets rather small rather quickly in most cases. (Field from each pole decreases as inverse square (not many people realise this)and the vector sum of the dipole pair approximates to inverse cubed at many magnet lengths away from the dipole center). 

A modern high strength rare earth magnet (usually Nd2Fe14B) will produce around 1 Tesla out to half of one magnet dipole (N-S) length from the pole face. ie long (or deep) magnet = deep external field. You can pretend that means it will be about 1/8th T at 1.5 magnet lengths and 1/27 Tesla at 2.5 magnet lengths etc. 



A MEMS accelerometer (probably) contains moving conductors and so may have some issues. You'd expect their data sheet to say so if this was important.

Any magnetic cored device that isn't shielded, and some that are shielded, could potentially be affected. For example a coil with a ferrite slug or one wound on a ferrite or iron core bobbin would have the AC BH curve moved by a DC offset value by the magnet's field and depending on magnet strength and proximity it could push a design into saturation or deeper into saturation than it otherwise would be. 

A magnetic style loudspeaker or earpiece could be affected. 

A Hall cell, GMR sensor, AMR sensor, and other explicitly magnetic field sensitive device 'could have fun'.

Any common mechanical meter movement could be affected (moving coil, moving iron, air core, ...)

Any electric motor (brushless DC, brushed, induction, stepper, head actuator, ...), relay or actuator using magnetic fields could be affected 

Maybe:

FRAM memory, core memory

Long bow:

Light Saber, Dilithium energy cell, ...



Should be OK:

As long as no specifically magnetic sensitive components - 

ICs, analog and digital, memory, RF (note inductor cores), ..
 Battery
 Passives - resistor, capacitor, ...
 Inductor, air cored.