Preciso de tirar fotocópias do formulário antes da reunião.

Because Portuguese often leaves out the subject pronoun when the verb already makes it clear who the subject is.

• Preciso already means I need
• So (Eu) preciso de tirar... and Preciso de tirar... both mean the same thing

In European Portuguese, omitting eu is very common unless you want emphasis or contrast.

In standard European Portuguese, precisar normally takes the preposition de before a noun or an infinitive.

Examples:

• Preciso de ajuda = I need help
• Preciso de sair = I need to leave
• Preciso de tirar fotocópias = I need to make photocopies

So in Portugal, preciso de tirar is the expected structure.
You may sometimes hear preciso fazer in other varieties, especially in Brazilian Portuguese, but preciso de is the safer European Portuguese pattern.

Yes, tirar often literally means to take or to remove, but in Portuguese it is also used in several idiomatic expressions.

One of them is:

• tirar fotocópias = to make photocopies

So although the literal idea may seem odd from an English point of view, this is a normal expression. In Portugal, tirar fotocópias sounds very natural.

You may also hear:

• fazer fotocópias = to make photocopies

Both are correct.

These are contractions:

• do = de + o
• da = de + a

So:

• do formulário = de o formulário
• da reunião = de a reunião

Portuguese normally contracts these combinations.

This also shows the gender of the nouns:

• o formulário = the form
• a reunião = the meeting

Here, do formulário means that the photocopies are copies of / from the form.

So the idea is:

• tirar fotocópias do formulário = make photocopies of the form

In English, we would usually say of the form, but Portuguese uses de here.

Because antes is followed by de, and when de comes before the article a, they contract:

• antes de + a reunião → antes da reunião

So:

• antes da reunião = before the meeting

If antes is followed by a verb, then you keep de without an article:

• antes de sair = before leaving
• antes de tirar as fotocópias = before making the photocopies

Because the speaker needs to make more than one photocopy.

• tirar uma fotocópia = make one photocopy
• tirar fotocópias = make photocopies

Using the plural here is very natural if several copies are needed.

If you wanted the singular, you could say:

• Preciso de tirar uma fotocópia do formulário antes da reunião.

Yes. Cópias is broader and can sound a bit more general, while fotocópias specifically means photocopies.

So these are both possible:

• tirar fotocópias do formulário
• tirar cópias do formulário

If you want to be very specific about using a photocopier, fotocópias is clearer.

Yes, it is natural in European Portuguese.

A few common alternatives are:

• Tenho de tirar fotocópias do formulário antes da reunião.
  = I have to make photocopies of the form before the meeting.
  This can sound a bit stronger, more like obligation.

• Preciso de fazer fotocópias do formulário antes da reunião.
  Also correct and natural.

• Preciso de copiar o formulário antes da reunião.
  This can work, but it may sound less specifically like using a photocopier.

So your original sentence is perfectly good.

In European Portuguese, small unstressed words often get reduced in pronunciation.

So preciso de may sound closer to:

• preciso d’
• or a very reduced de

The de is still there grammatically, even if it sounds weak.

This kind of reduction is very common in European Portuguese and is one reason spoken Portuguese can sound fast or compressed to learners.

Yes. Portuguese is flexible with adverbial phrases like antes da reunião.

For example:

• Preciso de tirar fotocópias do formulário antes da reunião.
• Antes da reunião, preciso de tirar fotocópias do formulário.

Both are correct.
The original order is neutral and natural: main message first, time phrase last.

The conjugated verb is:

• preciso = I need

Then comes the preposition:

• de

Then the infinitive phrase:

• tirar fotocópias do formulário = to make photocopies of the form

So the structure is:

• Preciso
  • de
    • tirar fotocópias do formulário
      • antes da reunião

This is a very common Portuguese pattern:

• Preciso de estudar
• Preciso de trabalhar
• Preciso de ligar-lhe