1. task left

You can use both; there is no difference between the two; they both mean essentielly that there is something that has not been done yet and should be done by the people on behalf of whom the person is talking. There exists however a very slight difference (can be ignored often) and that is in the judgement about the attitude towards doing what is left: in the first case it cannot be inferred from the statement that the person speaking should have a more or less compelling obligation to do what is left, or even that they are the ones to do it themselves, whereas as regards the second possibility there can be added the idea that they have a certain such responsability.

2. last ressource

In this case it is better to use the first possibility as it expresses specifically the notion of unique alternative; however a slight modification of the second possibility elicits its use in the way of expressing the same state of affairs; here is an example;

On a manqué le bus , il vient de passer et c’était le dernier… Nous n’avons plus que la solution de rentrer à pied!

« Nous n’avons plus qu’à rentrer à pied! » is not exact but possibly certain persons will occasionally use it and it is still understandable.

ADDITION 1

A point can be made to the effect that this form is feeble in this context; there is an insinuation of ambiguity that the context does not quite obliterate here; that is caused by another context of use (example of context: a man with the new task of cooking for the household).

• Je n’ai plus qu’à faire le ménage puis passer le reste de mon temps à regarder la télévision. (understated or expressed: before, that was all I had to do but now I’ve also got to cook.)

Of course, you’ll say “yes, but the context of the missed bus is totally different, the message communicated shouldn’t be confusing.” and it is difficult to render this argument ineffective.

Let’s mention however that “Il ne me reste plus qu’à faire le ménage.” can in no way thrive with this last “household” possibility, therefrom the greater specificity of the “ne reste plus que” option in the “missed bus” context.

END OF ADDITION 1

3. politeness

This case is very similar to the first except that instead of involving a statement about the conclusion of a chain of more or less programmed actions (which might consist of a single action) it is a statement about the conclusion of an event, the end of an event; in this case an important difference is to be taken into account, though; the first possibility makes for a statement implying only the appropriatness of the moment for the person to extend their farewell wishes; if the second posibility is used there is communicated a sense of obligation and that, although not the worst of mistakes, should be avoid, except, of course, if in a special circumstance the speaker wants to pass on those connotations; it is not however the normal situation, when one does better to keep to the use of “Il ne nous reste plus…”.

ADDITION 2

4. new obligation/necessity

For this case will only do the second form (avoir).

examples
– Nous n’avons plus qu’à nous présenter, donner l’adresse de l’ancient employeur et nous avons du travail tout de suite.

5. new minimal necessity

Only the second form (avoir) will do.

examples
– Vous n’avez plus qu’à vous rappeler ce numéro et vous êtes assuré d’une assistance 24 heures sur 24.
– Vous n’avez plus qu’à sortir une petite boite de votre poche et vous parlez pratiquement à n’importe qui n’importe où.