Does increasing kernel size in a CNN result in higher accuracy on the training set? In a convolutional neural network, does increasing the size of kernel always result in better training set accuracy? For example, if I use 5x5 kernels in a CNN instead of 3x3 ones, will it always generate better training accuracy? 

Increasing kernel size means effectively increasing the total number of parameters. So, it is expected that the model has a higher complexity to address a given problem. So it should perform better at least for a particular training set. Or will it be harder to learn a bigger kernel?

I am not concerned about the validation set accuracy here.
I'd say there is no direct relation between the kernel size and the accuracy.

If you start using larger kernel you may start loosing details in some smaller features (where 3x3 would detect them better) and in other cases, where your dataset has larger features the 5x5 may start detect features that 3x3 misses.

So, I'd say "no".

Anyway, if you add a second convolutional layer on top of your first you'd start having something that's close to larger features. E.g. your small kernels in the first layer would detect small features but the second layer would detect features which are composed of several features from the previous layer.