Ttgjyuk

Do this to avoid having to do the dishes.

Do this to not do the dishes.

I am not sure what tense the first sentence is, I feel it's in the simple present too, but I am not sure. Is there a temporal difference in meaning between the two? I feel the difference is that the first one implies doing the dishes is an obligation, but there doesn't seem to be a "temporal difference in meaning".

It's simple present imperative, just like the second. They just have the adverbial of purpose ("to...") phrased differently.

The first uses a catenative, where a verb takes another verb or verb phrase as its argument. Avoid is catenative, with having to do the dishes as its argument. To have here can also be described as a catenative verb, but it's easier to just think of it as a modal auxiliary showing obligation.

As a catenative, avoid takes a gerund or gerund phrase, representing the action that you are avoiding. That action is having to do the dishes. Thus, it is avoiding the obligation to do the dishes. You could simplify it by removing the explicit obligation:

Do this to avoid doing the dishes.

In all of these versions, the principal verb is the first do, and the other verbs - avoid, having, and the second do (whether it's negated or not) - are in non-finite forms, and have nothing to indicate time. Non-finite verbs can have indications of time, but often they do not. Thus, there is no difference in time between them - one simple expresses, with clarity, that doing 'this' will allow you to avoid the obligation to do the dishes. The other says that doing 'this' will allow you to not do the dishes. The eventual meaning is the same, the difference is there but is not of practical importance, and there is no difference in time between the two.

** EDIT **

I see SamBC answered before me, and his answer is probably better and more accurate. Just adding this in case it includes anything of use.

** END EDIT **

"Have to" is a fixed phrase meaning 'to be obligated [to do something in the infinitive]', so it is easiest to think of it as if it were a single verb rather than two words, when parsing.

Consider this phrase:

Why am I doing the dishes? Because I have to.

For an English learner, it probably looks completely wrong. In effect the sentence is being ended by a floating auxiliary 'to' from a verb which isn't even included ('do' is being implied). But it is a very common English construction.

In this case the verb "have to do" is in the gerund form. Why? Because it is following a verb which can be (and commonly is) followed by the gerund form.

I avoid running for the bus.

He wants to avoid doing his English grammar homework.

Therefore, these sentences are equally correct:

We all avoided having to do the dishes.

You never avoid having to run for the bus.

The tense of the sentence depends on the first verb: avoid. If avoid is present tense, then the sentence is present. If you said:

I avoided having to do the dishes.

then it would be past. In your sentence you have the present infinitive.

Do this to avoid having to do the dishes.

Do this to not do the dishes.

No, they are not simple present. These are both imperative clauses, which use the plain form of the verb and thus are tenseless, though finite.

They both have embedded infinitival clauses functioning as purpose adjuncts, the first containing a further embedded gerund-participial clause as catenative complement of "avoid", which in turn contains a further embedded infinitival clause as catenative complement of "having".

The second is quite unnatural. Stick with the first.