If I undergo suffering now, will my next birth be better?

Karma is of course accumulated. But other factors play much more important role in it's ripening than exhausting of accumulations. Our actions in this life and (impossible to control) thoughts at the last moments of life are usually much more determinant to next birth than expiration of karma or merit. See quotation from Abhidharmasamuccaya:

A being is accompanied by many actions. In this case, what is the order of results of maturation?

i) First to produce its results is the most serious action;¹⁹⁰

ii) next, the action that appears at the moment of death;¹⁹¹

iii) then, that which was habitually done previously;¹⁹²

iv) finally, that which was done earlier in a past lifetime.¹⁹³

(footnotes):

[190] Pradhan’s restoration: audārika-karma. But very probably guru-karma. Cf. Kośa, IX, 297. In Pali also it is garu- or garuka-kamma. [191] Āsanna- or maraṇāsanna-kamma in Pali. [192] In Pali ācinnaka- or bahula-kamma. [193] This refers to kaṭakka-kamma in Pali . These four actions produce their results in this order. Serious action has priority over the others. If there is no serious action, then an action which presents itself to the mind at the moment of death produces its results. If no particular action presents itself to the mind at the moment of death, then the action most frequently done produces its results. If none of these three actions has the opportunity of bearing results, then an action done in a past lifetime will produce its results. See Kośa, IX, 297.

To rely on expiration of karma is probably even wrong view, because it is conductive to lack of right effort to do wholesome actions that actually improve karma.

Also, without right effort you'll continue to produce bad karma so there is no exhaust in this case.

This from Bodhidharma

Every suffering is a buddha-seed, because suffering impels mortals to seek wisdom. But you can only say that suffering gives rise to Buddhahood. You can’t say that suffering is Buddhahood. Your body and mind are the field. Suffering is the seed, wisdom the sprout, and Buddhahood the grain. The Buddha in the mind is like a fragrance in a tree. The Buddha comes from a mind free of suffering, just as a fragrance comes from a tree free of decay. There’s no fragrance without the tree and no Buddha without the mind. If there’s a fragrance without a tree, it’s a different fragrance. If there’s a Buddha without your mind, it’s a different Buddha

Emphasis mine.

Given the conventional theory of karma (the actor is not different to who experiences the result) it seems like you're asking whether bad karma can ever be exhuasted. I would say yes. There's definitely an imperative to avoid both extremes (austerities and indulgence), though, even if that only applies to those on the path, which it likely doesn't anyway

If an individual is suffering because they are overcoming the five hindrances than probably. If a person is suffering because they are indulging in the five hindrances than probably not.

If I eat a big plate of my favorite food every night I will suffer from it because there will be times when I am not eating it, there are times I will feel sick from too much of it and many other ills associated with the indulgence. This suffering (unless it leads to the eventual cessation of desire in regards to this food) will not help my future births.

If I am addicted to eating my favorite food every night and I realize how foolish it is to indulge in it, and I decided to stop, I will suffer during the process of overcoming this attachment. This suffering will help my future births.

In regards to OP's comment: A person who recently got a disease would not have any more inherent suffering in their life than someone who did not have a disease given that they are clear of the hindrances.

The suffering is compounded when the individual desires feeling good and healthy (sensory desire), or is worried about death or further illness (worry), or is angry that they are sick when someone else they know deserves it more (ill-will), etc, etc.... That type of suffering, the same as in the example regarding food, will not help; overcoming attachments to these hindrances can incur a lot of suffering (I am not my good health, I am not worried about "death", no one deserves anything, etc, etc...; respectively) and that suffering can help future rebirths.

Since there are great answers already, I won't be answering. So consider this not as an answer but as an appendix that might help you discern the details of the answers.

• MN 14, The Shorter Discourse on the Mass of Suffering (Cūḷadukkhakkhandha-suttaṃ)
• MN 45, The Shorter Discourse on Ways of Undertaking Things (Cūḷadhammasamādāna-suttaṃ) ^*
• MN 46, The Greater Discourse on Ways of Undertaking Things (Mahādhammasamādāna-suttaṃ) ^*
• MN 101, To Devadaha (Devadaha-suttaṃ)

* These discourses seem to answer the question directly.

Devadaha Sutta (MN 101) linked to by @Yuttadhamo is the right sutra to consider, that fully covers this topic. As you can see in MN 101, the Jains too held position that "burning" past karma through exertion and austerity can lead to liberation.

As he interrogates Jains, Buddha makes following points:

• Not all pain comes from causes in the past. Some pain obviously comes from fierce striving, fierce exertion performed in the here-and-now. (As my first teacher used to say, sometimes pain comes from dental crowns and not from karma.)
• Unfortunately, we can't really see how much negative karma we have accumulated, and how much karma has been exhausted so far.
• If we saw that, then in theory we could talk about removing karma through exertion, similarly to how surgeon removes a poisoned arrow, producing pain short-term, but stopping it long-term.
• Unfortunately, that is not possible, because:
  • Unfruitted karma that is to fruit in one way, cannot be converted to fruit in another way, through striving & exertion.
  • Unfruitted karma that is to fruit in the future, cannot be converted to fruit in the present, and vice versa, through striving & exertion.
  • Unfruitted karma that is to bring result of one size, cannot be converted to bring result of another size, through striving & exertion.

Instead, according to Buddha's teaching in MN 101, liberation from dukkha is achieved by "abandoning of unskillful mental qualities and the attainment of skillful mental qualities in the here-&-now".

Specifically, "skillful mental qualities" refer to 1) "exertion of fabrication against a cause of dukkha" which leads to "dispassion due to the fabrication", and to 2) "looking with equanimity at the cause of dukkha", which leads to "dispassion through the development of equanimity". In other words, cessation of dukkha is attained through 1) deliberate control of one's perspective, and through 2) equanimity that comes from philosophical attitude.

As an example of typical dukkha-pattern, Buddha tells a story of unshared love, with the protagonist suffering from "sorrow, lamentation, pain, distress, & despair" due to being "in love with her, his mind ensnared with fierce desire, fierce passion". The protagonist then wilfully abandons "desire & passion for that woman" (an effort that serves as "exertion of fabrication against a cause of dukkha") and as a result attains dispassion "due to the fabrication".

With all of the above in mind, Buddha still admits to importance of pushing oneself and overcoming one's limits -- to avoid regress due to indulgence -- but only inasmuch as to have "unskillful qualities decline" and "skillful qualities increase":

So he exerts himself with stress & pain, and while he is exerting himself with stress & pain, unskillful qualities decline in him, & skillful qualities increase. Then at a later time he would no longer exert himself with stress & pain. Why is that? Because he has attained the goal for which he was exerting himself with stress & pain.

As a result,

A monk, when not loaded down, does not load himself down with pain, nor does he reject pleasure that accords with the Dhamma, although he is not fixated on that pleasure.

I never really understood this view, but it seems to have been held by the Niganthas. See MN 101 for a detailed argument against this sort of reasoning (too long to quote here).

Basically, though, it is like saying that if I have taken poison, I should take more poison so I feel the poisoning more quickly and thus somehow work out the result of the original poison quicker. Isn't that a ridiculous idea?

If I have performed karma X in the past that has result A in the future, performing other karmas Y and Z that also have result A doesn't do anything (except perhaps coincidentally - e.g. cutting off my arm means I can't experience pain in my hand) to affect karma X.

Another fallacy here is the idea that experiencing suffering leads to future happiness (in a way that pleasure doesn't). Really, to be objective, your question would have to include pleasure, in which case it would be, "If I undergo pleasure and pain in this life, will my next rebirth be better?" to which the answer is of course no.

An important concept to always keep in mind, according to the Buddha's teaching, is that pleasure and pain don't lead to pleasure or pain, it is good and evil that lead to pleasure and pain.