First, let's define 'diet soda' as any drink that is sweetened with low to no calorie sweeteners (like sucralose, aspartame, acesulfame-potassium, etc.) in order to taste like regular (sugared) soda.
In short, there are no direct causative studies that show the use of artificial sweeteners slows your metabolism or promotes fat storage.
The CHOICE randomized study, lasting 6 months and observing how different diet changes affect weight loss adherence, found that diet soda is an effective substitution for regular soda and had no significant differences in health or weight loss relative to water.
Data from the NHANES (03-04) study suggest that increased servings of non-caloric beverages are not related to an increase in total calories. Additionally, the PREMIER trial found that a reduction in intake of non-caloric beverage is not associated with weight loss.
The real problem is that people tend to under-estimate caloric intake. By thinking about the zero calories in the soda, they end up eating an excess of calories, thus increasing fat storage. This makes the well known correlation (relationship) between diet soda and obesity; one does not cause the other, but they tend to co-exist. Drinking diet soda does not make one fat, but people with obesity tend to drink diet soda.
In regards to insulin, diet soda can potentially increase insulin secretion by both an anticipatory response and the artificial sweetener aspartame can (through the amino acid, phenylalanine). However, both of these insulin spikes are too small to matter practically, and the latter mechanism doesn't seem to occur at all with many commonly ingested dosages.
As mentioned on our Aspartame and Appetite page, aspartame (via phenylalanine) might also suppress appetite.