Learn Hardcore Russian: Не стоит увеличивать бюджет проекта без причины. - It’s not worth increasing the project budget for no reason.

This is an impersonal construction. (Не) стоит + infinitive expresses general advice/assessment: it’s (not) worth / it’s (not) a good idea to…
Because it’s impersonal, Russian doesn’t need an explicit subject like you/we/one. The “doer” is understood from context, or it’s meant to be general.

What exactly does не стоит + infinitive mean, and how strong is it?

Не стоит + infinitive usually means it’s not worth doing / you shouldn’t (really) do it / it’s not advisable.
It’s often softer than a direct command like не увеличивайте and a bit more conversational than не следует.

Why is the verb увеличивать (imperfective) used instead of увеличить (perfective)?

With не стоит, Russian commonly uses the imperfective infinitive to talk about an action in general (as a practice or idea).
Не стоит увеличивать бюджет… = don’t make a habit of increasing it / it’s not a good idea to increase it (in general).
You can sometimes see не стоит увеличить…, but it tends to sound more like a one-time, specific action in a particular situation; many speakers still prefer imperfective after не стоит.

Is не стоит literally “doesn’t cost,” like стоить usually means?

Same verb, but here it’s a different common meaning of стоить: to be worth (doing).
So это стоит сделать = it’s worth doing; не стоит делать = it’s not worth doing / you shouldn’t do it.

Why is it бюджет проекта and not бюджет проект or something with для?

проекта is genitive singular: the project’s budget / the budget of the project. This “noun + genitive” pattern is extremely common for possession/association.
You could also say бюджет для проекта, but that shifts meaning toward a budget intended for a project, not necessarily “the project’s budget” as an established entity.

What case is used after без, and why is it без причины?

без requires the genitive case.

nominative: причина
genitive: причины
So grammatically it’s без причины = without a reason.

What’s the difference between без причины and без причин?

Both are possible, with a nuance:

без причины (singular) often means without any (good) reason in a general sense.
без причин (plural) can emphasize with no reasons at all / “without grounds,” sometimes sounding a bit more categorical or formal.
In many contexts they’re interchangeable.

Could the word order change? For example, Без причины не стоит увеличивать бюджет проекта?

Yes. Word order is flexible. Moving без причины to the front gives it more emphasis (topic/contrast):

Не стоит увеличивать бюджет проекта без причины. (neutral)
Без причины не стоит увеличивать бюджет проекта. (emphasizes without a reason)
The meaning stays basically the same; emphasis changes.

Is проекта pronounced with stress at the end? How do I stress the whole sentence?

Common stresses: не сто́ит увеличи́вать бюдже́т прое́кта без причи́ны.

сто́ит: stress on о
увеличи́вать: stress on и
бюдже́т: stress on е
прое́кта: stress on е́
причи́ны: stress on и́

Are there close alternatives to не стоит that sound more formal or more direct?

Yes, common alternatives:

Не следует увеличивать бюджет проекта без причины. (more formal/official: should not)
Не нужно увеличивать… (practical/neutral: no need to)
Не увеличивайте… (direct instruction/command)
Нет смысла увеличивать… (emphasizes lack of point: there’s no point in increasing…)