would tire size 195/65/R15 affect car transmission?

Here's the reality. The 195/65-R15 tires are well within the acceptable range for your stock rims and wheel well clearances. The tires are 0.4mm wider and 0.6mm taller than the stock tires. That means your odometer and speedometer will be 2.3% slow. For example, your speedometer will say 60mph when your true speed is 61.4mph. Your Toyota dealer can easily recalibrate your car to account for the difference, if you desire. The 195/65-R15 tires will do nothing to your transmission or cause any other mechanical failure. A slight increase in tire width may offer slightly improved grip. A slight increase in tire height may slightly decrease handling and improve ride (adds floatiness) due to the taller sidewall. Using those tires will not affect your warranty whatsoever, since Toyota attempting to do so would violate the Magnuson-Moss Warranty Act.

You would have to do something that significantly affects vehicle performance and reliability to risk voiding related aspects of your warranty, e.g. aftermarket suspension modifications, ridiculously undersized/oversized aftermarket tires, very heavy aftermarket wheels, add a turbo, radically reprogram your engine's computer settings, etc. Don't sweat it.

If you don't own the 195/65-R15 tires yet, 195/60-R15 is a better choice at 0.4mm wider but 0.2mm lower profile, which should give better grip and handling.

 

I have used a set of 195/65-R15 Blizzacks on my 05 Prius for the past 2 winters with no problems. Had the car in for service at the dealer and received no comments from the mechanics. Go with the winter tires and have a safer winter!

 

I suggest you go with 205/60R15 like I did. The tires are nearly same diameter as the OE ones (185/65R15). A 205 tire is 2cm wider than a 185 (unit is 1/10 cm=1mm). Keep in mind that the OE Goodyear Integrity is one of the worse tires performance-wise. I have been running 205/60R15 Sumitomo. So far I have been seeing 2-3mpg drop, but an increase to 35mph over 31mph on the same 270 degree down ramp near my house. A 12% increase in roadholding g-force is not "little" or "slight" improvement in my book. Will you trade 3mpg for it? Is it up to you.

Definitely no problem with tranny. Your mechanic is smoking something funny. I have sized up tires on all my vehicles (plus zero or plus one). I have never got warned by any dealers or menchanics. Try to keep the same diameters (1/8" to 1/4" difference is acceptable), though.