In the sentence "I watched the balloon rise", "rise" is in what verb tense? And why?

Other examples of this structure or form.

• He saw the man leave.
• They heard the door bang.
• She felt her pulse race.
• We watched the house burn.
• I listened to the music play.
• He looked at the squirrel run.
• We experienced the situation worsen.
• I perceived the the bell ring.
• We noticed the door open.

So, those are some of the verbs that can work this way. They all involve having a direct experience of some thing.

If you take the direct object and replace the present simple verb with when it + the verb, the meaning is clear.

• I saw the man leave. = I saw the man when he left.
• They heard the door bang. = They heard the door when it banged.

The examples above where the third person plural s is not used, are just like:

• We suggest you leave.
• I recommend he go.
• It is important he play today.

However, in the first group of examples, the main verbs are all in the simple past. Except for stage or movie script directions, the simple present here would be odd, though the continuous would not:

• He sees the man leave. [theatrical script]
• He saw the man leave. [real life]
• I'm seeing the man leave. [real life, as he leaves]

This used to be called subjunctive in English. Many linguists would call this a bare infinitive today.

Personally, I would argue that these examples parse like this:

• He saw//the man leave

And that "the man leave" is simply a direct object phrase in the form of a subject plus bare infinitive as with verbs like recommend or advise or suggest.

Same as: I watched//the balloon rise.

These can also be analyzed as noun clauses:

For example, from the linked text:

The sales clerk suggested she put the dress on hold.

noun clauses

To summarize, some verbs of observing and experiencing function in a way similar to the verbs advise, recommend and suggest where the noun phrase verb in the third person singular does not take an s.

I watched the balloon rise.

"Watch" is one of a few verbs of sensory perception that can take a bare infinitival complement, but not a to-infinitival complement.

Most of the other verbs of sensory perception can take both a bare and to-infinitival complements, cf. I saw him leave and the passive He was seen to leave.

There are plenty of resources on the 'Net, where you can find more information about other verbs whose complements are restricted to one or the other, or that can take both kinds of infinitival complement.