One of the biggest problems I can think of is that you become very likely to generate samples that do not exist in the data you generated your model from.
An example could be if you made a model that generates mammals based on their features such as whether they have a tail, how many eyes they have and the number of legs.
Most mammals have 4 legs, so let's assign the probability 0.9 to that feature and a small subset (humans and monkeys I guess) have 2 legs, so they get a probability of 0.1 and the probability for any other number of legs is 0.0.
As far as I know there are no mammals with more or less than 2 eyes, so let's just assign the probability 1.0 to that and 0.0 to everything else.
1 | 2 | 3 | 4 | 5 |...
Now, if you are maximizing over the product of probabilities you will find that the probability of generating an animal with 2 eyes and 4 legs is 0.9, 2 eyes and 2 legs is 0.1 whereas generating an animal with 2 eyes and 10 legs is 0.0, so you're staying pretty close to observations you would see in your training set.
p(legs=4, eyes=2) = 0.9\\
p(legs=2, eyes=2) = 0.1\\
p(legs=5, eyes=2) = 0.0
If you instead maximize over the sum of probabilities a mammal with 2 eyes and 4 legs will get a value of 1.9, 2 eyes and 2 legs 1.1 and a mammal with 2 eyes and 3 or even 400 legs gets a value of 1.0.
f(x) = \sum p(x)\\
f(legs=4, eyes=2) = 1.9\\
f(legs=2, eyes=2) = 1.1\\
f(legs=5, eyes=2) = 1.0
The problems here are of course that first of all these values can no longer be interpreted as probabilities as they are no longer constrained to [0, 1], but also that samples not observed in the training set are almost as likely as observed but rare samples.
Maybe there are cases where generating one probable feature makes outlandish values for other features likely, but nothing immediately springs to mind.