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`～おう` means "let's do something", while てしまう seems to mean completion of an
action (A dictionary of basic japanese grammar). In Nihongo so-matome n3 there
is this example: `今日は飲んじゃおう!` which they translate to `I'm going to have a
drink!`. What exactly does ～てしまおう mean?

I've seen some google results for this question, but they were on reddit and
few other websites I can't access from this network.
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て-form
What does しまおう mean?
Just to clarify,「〜おう」is called the volitional form, and though it is often
translated as "Let's do ~", it is also used to express individual intention.
When used in the individual context, as far as I understand it, it expresses a
strong intention to do/achieve something. You'll often hear it used in male
speech and followed by 「ぜ」.

Example: やろうぜ！ I'll do it (for sure)!

Combined with 「〜てしまう」it basically means going all out or doing something
fully. Here's another example:

私は何もかも言ってしまおう。I will let myself loose.

Another clarification: while しまおう and じゃおう mean the same thing, they are not
interchangeable. They are dependent on the て form that a particular verb
takes. For example, 言う becomes 言ってしまおう becomes 言っちゃおう, but because 飲む becomes
飲んで, and で is a softer sound than て, the conjugation is also softer, hence it
becomes 飲んじゃおう, not 飲んちゃおう. However, when using the more polite form, it
remains 飲んでしまう, not 飲んでじまう. Hope that helps.
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The te-form is used for stringing verbs together to combine their meanings:

> 飲[む]{L} + しまう→飲[んで]{LL} + しまう→飲んでしまう

しまう usually means "to complete something", but what it actually means depends
on context. [In an answer to an earlier
question](https://japanese.stackexchange.com/questions/36672/v%E3%81%A6%E3%81%97%E3%81%BE%E3%81%86-%E4%BB%96%E3%81%AE%E4%BD%BF%E3%81%84%E6%96%B9%E3%81%8C%E3%81%82%E3%82%8B),
its combined meaning with 飲む is explained as "drinking without thinking about
consequences". An example of 〜しまう meaning "to complete something" (from
naruto's comment):

> 賞味期限が明日だから、今日飲んでしまおう  
>  _It'll expire soon, so better drink it all today_

The volitional form is used to mean things like will, invitation, doubt or
presumption. Here, しまう is conjugated to the volitional form and because the
sentence starts 「今日は」, you can guess that it means the speaker's will to do
something.

> 飲んでしま[う]{L}→飲んでしま[おう]{LL}

If you view 飲んでしまう as "drink unresponsibly", then 飲んでしまおう will become "I'll
drink unresponsibly/anyway".

For comparison, volitional without 〜しまう:

> 飲[む]{L}→飲[もう]{LL} (I'll drink, let's drink)

In Japanese, te-form + aux is often contracted when it connects to frequently
used auxiliary verbs like 〜しまう or 〜いる. 飲ん[でしま]{LLL}う becomes 飲ん[じゃ]{LL}う and
言って[い]{L}る becomes 言ってる. The contracted form conjugates to the volitional form
just like the longer one.

> 飲ん[でしま]{LL}おう↔飲ん[じゃ]{LL}おう
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First, ～てしまう doesn't so much mean completion as to do something in an
unrewindable manner.

Second, volitional forms (～おう) stands for invitation or volition, in other
words, it means either "let's do it" or "I'll do it". (When it's accompanied
with sentence ender よ or ぜ, it's invitation only.)

Combination of them expresses encouragement for decision against some
hesitation. As for your example, 今日は飲んでしまおう implies that you shouldn't care of
extra problems that could happen after it.
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