Feeling of forcing oneself to do something

Perhaps not exactly what you want, but worth keeping in mind…

constancy

1 a : steadfastness of mind under duress : fortitude
   b : fidelity, loyalty
                                           (Merriam-Webster)

Sometimes people do things out of a sense of duty, so a daughter might dutifully tend to her elderly mother even though the motivation for so doing has long gone.

She goes through the motions because she feels she ought to.

As Lily rightly points out, "duty-bound" introduces the sense of being forced into doing something. Well spotted, Lily.

strain

He loved playing baseball at a young age. He never missed a practice. By the time he was 18, though, he had lost most of his interest in sports, and going to practice was a strain.

push yourself

He loved playing baseball at a young age. He never missed a practice. By the time he was 18, though, he had lost most of his interest in sports, and had to push himself to go to practice.

strain

noun: severe, trying, or fatiguing pressure or exertion

dictionary.com

I would suggest resolve or resolution depending on the phrasing employed at the time.

So, given an unattractive task that has to be undertaken sooner or later, you would resolve to get on with it, and once engaged, continue with resolution, despite a desire to abandon the damn thing and go down the pub.

Grit. Merriam-Webster represents a common meaning:

: firmness of mind or spirit : unyielding courage in the face of hardship or danger

Recently, the term has been applied in education and psychology to refer to people who can stick with challenging tasks and persevere through boredom and disappointment to attain a larger goal. In the article "Grit: perseverance and passion for long-term goals" (Angela L. Duckworth, Christopher Peterson, Michael D. Matthews, Dennis R. Kelly, Journal of Personality and Social Psychology, 2007), they identify perseverence through disappointment and boredom as a key quality of grit:

We define grit as perseverance and passion for long-term goals. Grit entails working strenuously toward challenges, maintaining effort and interest over years despite failure, adversity, and plateaus in progress. The gritty individual approaches achievement as a marathon; his or her advantage is stamina. Whereas disappointment or boredom signals to others that it is time to change trajectory and cut losses, the gritty individual stays the course.

The original article has been cited by over 4000 other texts (Google Scholar), spawned a 2016 book by the original primary author of the study, and became a buzzword (sometimes criticized) in education. It has filtered extensively into public-facing websites, like this one from Steyning Grammar School in the UK:

Grit is resilience to challenges accompanied by perseverance and passion for long term goals.

We want our students to be 'gritty' and when they are we notice that they:

• 'Come back' after failure or low grades
• Don't give up easily when they do tough tasks
• They finish longer tasks with a high quality outcome
• Can focus on the fact that what they do now affects their GCSEs or other long term goals

Sounds like reluctance to me.

: feeling or showing aversion, hesitation, or unwillingness

: having or assuming a specified role unwillingly

With what you've given I feel like "duty" is probably the best fit.

Duty something that one is expected or required to do by moral or legal obligation

https://www.dictionary.com/browse/duty

Edit: added source per suggestion

It's not a word, but a phrase I might use to describe getting up the wherewithal to do something I don't want to do but I have to do it is: gird my loins

It means preparing yourself for something that's coming up, usually dangerous or difficult.

E.g.:

I don't want to do the washing up, but it needs doing, so I'm girding my loins to do it.

It doesn't exactly mean doing the thing, but it indicates that the thing is something you don't want to do but have to do anyway.

perfunctory

(of an action) carried out without real interest, feeling, or effort. "he gave a perfunctory nod"

https://www.merriam-webster.com/dictionary/perfunctory

One can have a sense of obligation to do something.

This is related to the "sense of duty" other answers discuss, but I just wanted to mention.

Examples:

Despite having no motivation to continue in the race, John felt an obligation to compete to show his family he could persevere and win.

There are so many people supporting me in my journey, despite my lack of motivation, that I felt obligated to follow through for them.

Fiduciary Responsibility could also apply. It has a specific "business meaning" related to duty-bound, but could be easily borrowed. "Despite having long since lost the personal joy in competition, John felt a fiduciary responsibility to his teammates forcing him to continue showing up for games and playing them seriously."

However, my pedantic nature forces me to call out the fact that you have kind of asked an impossible to answer question. On a certain level, everyone is always motivated to be doing exactly what they are doing at every moment. Ultimately, there is no difference between 'internal' and 'external' motivation. Even under extreme externally caused duress, one's motivation for how to respond is always personal. We don't get to control what we encounter on our Path. We only get to control how we respond to such encounters.

Sounds like deign, but it does come with some additional baggage.

https://www.merriam-webster.com/dictionary/deign

"to condescend reluctantly and with a strong sense of the affront to one's superiority that is involved"

"to condescend to give or offer"

To "resign oneself" to some fact or necessary choice is to accept it. Also see "resignation".

It fits your sentence very nicely:

Jon resigned himself to continue participating in competition X, despite no longer having the genuine motivation for it

If what you really want is a verb that is the opposite of "procrastinate" in the way that you mean, or a noun that is the opposite of "procrastination", then I think you are out of luck. "Industriousness" doesn't quite fit for the noun because it it specifies a propensity, not the behavior itself. The best answers I would give are:

The way to describe forcing yourself to do something is "forcing yourself to do something".

The closest verb phrase to the opposite of procrastinating is something like "getting on with it." Maybe that's more British, but I'm an American, so I guess it's OK here too.

There are a host of metaphoric ways to get this idea across. "She gritted her teeth and began to work."

I'm looking for a word that captures that feeling of forcing oneself to do something that is not motivating anymore.

I believe a word that captures this feeling is ploddingly. The definition of plod is to work laboriously (with difficulty/effort) and monotonously (in a way that does not produce interest). As an adverb, ploddingly would mean "characterized by working in a difficult and uninteresting manner". We can apply this to your requested sentences with modifications/improvements:

Jon ploddingly continues to participate in competition X, despite losing his motivation for participation. Apparently, he values being a pacesetter enough to take up otherwise meaningless pursuits over a hobby.

In these sentences, the connotation is that Jon feels that continuing to participate in competition X is an uninteresting burden, but that he continues to participate anyways.

This might not be entirely grammatically correct, but you could say that "John grinds out his homework/sports drills/etc.". It's a bit awkward but I think it's quite close to the meaning you want.