How do different kinds of motors behave electrically under operation?
There are too many variables and motor ypes to do a good job of answering this universally without an extremely large answer.

Appliance motors are usually given a design life based on expected usage and desired longevity. eg automotive windscreen wipers may be designed for say 10% of vehicle operating hours. The following figures are "out of my head by way of example". Feel free to substitute your own assumptions. If target vehicle usage is 15000 km/year at average 50 km/hour that's 15000/50 = 300 hours so wiper usage at 10% = 30 hours so a 10 year lifetime = 300 hours wiper motor design life. You may decide to double or treble that if you don't want your vehicles to be known for minor parts failures as they age or may leave it as is if you value the aftermarket sales. Users who live in high rain areas may "have problems".

Your toothbrush motor will be built to meet a quite short operating life. As it is a brushed motor you have commutator wear and brush wear and bush holders need to last in the high vibration environment - but not cost more than can be helped. The windings may be held in place to some extent with an adhesive which will heat age. Operation at excees voltage will cause substantial extra I^2R heating and may cause windings to disintegrate much earlier than designed. 

In larger motors especially but quite possibly in your small one core iron will be designed to be on the top end of the BH magnetisation curve so that the steel is efficiently used. Excess voltage that pushes the core amp turns up by say 10% may result in very substantial core saturation, loss of magnetic field and high current leading to extra amp-turns leading in course to ... . .  

Bearings will be deigned for a certain operating life and higher revs may cause more heat, and may fling lubricant off bearing surfaces more readily and may discover resonances which impact motor life but which are not encountered at lower revs. 

In the case of your small motors, increase in resistance with use may be due to increased commuator to brush resistance as surfaces become oxidised uneven comm wear changes available surface area,brush tension may drop as brushes shorten and so spring pressure decreases and springs themselves may lose tension. 

Or not :-).
 ie these are some of the possible factors.
 How relevant each is needs to be determined in practice.