When referring to the total calories contained in the fruit -

The amount of calories cannot increase (ie. that would mean the banana, or whatever fruit you have in mind, is spontaneously gaining energy. Which we all know is impossible). That being said, it is very possible that the fruit is losing calories. When some fruit begin to ripen they give off chemicals that cue other fruit to begin the ripening process. These kinds of fruit are known as "climacteric fruits". Anyways, the chemicals being released, mainly ethylene, need to come from somewhere and when given off, the fruit "loses" a part of its self and thus the calories decrease, albeit by a very tiny amount.

When referring to the total number of calories we can remove from the fruit with our bodies -

When a fruit ripens it converts its starch reserves into sugar, hence the sweeter taste. Humans can digest both starch and sugar quite easily, therefore, the # of calories we can digest stays the same whether in starch form or in sugar form.

So, essentially, despite the fruit tasting sweeter when it ripens you are still getting the same number of calories... It's just that those calories reside in a form that we sense as "sweet". And if anything is changing in terms of calories, it would be a decrease, not an increase.

EDIT 1: Yes the nutritional information would be different. Mainly you are decreasing more complex carbohydrates (starch) for simpler carbohydrates (sugar). That being said... Our bodies are pretty damn efficient at converting starch to its individual monomer units, so there probably isn't any noticeable difference from our body's perspective.

EDIT 2 Okay so I busted out my old plant physiology textbook to clairify a few things, and hopefully provide some answers to some of the speculation posted in response to my comment. It's also worth noting that I will not be talking about the total calories contained the the fruit (since we have already established that this cannot change unless energy is physically absorbed) but rather, the bioavaliable calories, ie. the calories we absorb from the fruit. Furthermore, I will only be pointing out the changes in bioavailable calories that are relevant to us in terms of nutritional value (ie. no "there was a loss of 1 micro-calorie due to ethylene gas being given off").

Below is a summary of the four main effects of ripening on a fruit and what they mean in terms of changes in bioavailable calories

  • Starch hydrolysis: This is the process I was referring to previously where the starch is broken down and stored as sugar.

    Bioavailable calories: No net change

  • Chlorophyll degradation: Although I'm not positive what the chlorophyll is degraded into (about 70% sure it just forms other pigments), this is what is primarily responsible for the colour change seen in ripening. Also, chlorophyll and its derivatives are very much digestable, thus no change in bioavailable calories is seen here.

    Bioavailable calories: No net change

  • Organic acids & oils and phenolic compounds are metabolized: You may have heard of the word "Tannin" before. Tannins are phenolic compounds responsible for that "bitter" taste in unripe fruit (fun fact! Tannins are used as flavour additives in beer, wine, and tea!). The breakdown of Tannins has a two-fold effect: 1. Reduction in the bitterness of the fruit and subsequent "unmasking" of the sweet taste. 2 A net increase OR a net decrease in bioavailable nutirents... wtf right? Let me explain. Tannins may have an "anti-nutritional" effect. What does that mean? Anti-nutrients are compounds that interefere and disrupt the normal absorption of nutrients in our digestion (eg. caffiene). Why is it that tannins "may" have an anti-nutritional function? Well, the word "tannin" really refers to a class of bitter molecules found in plants, and as such, the type of tannin and amount varies between species. As it turns out, some tannins are anti-nutritional while others are beneficial to our nutrition, therefore, the breakdown of tannins during fruit ripening can either increase the bioavailable calories and nutrients (by eliminating tannins that have an anti-nutritional effect) or decrease the total nutritional value (by eliminating tannins that have a positive nutritional effect)... so it really depends on the type of fruit that is ripening*.

    Organic acids and oils are a different class than tannins, and make up tons of different compounds throughout the plant. Due to the sheer number and different effects of said compounds, I can't really comment on their digestability before fruit ripening. What I can tell you however, is that some of these compounds are degraded into sugar monomers during the ripening process, so depending on their digestability before degredation this could either mean an increase in available nutrients (if they were indigestable before) or no net change in avaliable nutrients (if they were digestable before).

    Bioavailable calories: Changes in bioavailable calories and nutrients will fluctuate depending on the type of fruit*.

  • Enzymatic breakdown of pectin: Pectin is the principle component of the middle lamella of the cell wall (for those of you unfamilier with plant anatomy just understand that plant cells are surrounded by walls, these walls provide the plant with a rigid support system, and that pectin is responsible for holding these walls together). Since pectin plays a crucial role in maintaining the rigidness of the plant, the breakdown of this compound causes the plant to "soften". The same logic applies to fruits, and thus provides us with an explaniation as to why fruits soften when they ripen. As it turns out, pectin is broken down into sugars which we can digest, however, I'm pretty sure (maybe somone can correct me if I'm wrong) that we can digest pectin, so despite it being broken down into sugars, we shouldnt see any increase or decrease in bioavailable calories.

    Bioavailable calories: No net change

As you can see, the total number of bioavailable calories stays pretty much constant through most effects of ripening, with the only noticable effect being the loss of tannin, which can either increase the total nutritional value or decrease it *see below.

*In my personal opinion, and coming from an evolutionary point of view, it is not beneficial for plants to have their fruit eaten before their seeds are ready. When the seeds are ready, the ripening process begins and tannins are broken down. It's not too far of a stretch to think that the tannins in unripe fruit would discourage consumption by both taste and by acting as anti-nutrients. Based on this thinking, I personally believe that the tannins in fruit are anti-nutritional and by eliminating them in the ripening process there is a net increase in bioavailable nutrients.