Can I travel to schengen areas with only my permesso di soggiorno?

It's not clear from the French government site how critical it would be for you to have your passport if your identity is checked by the police. As a foreigner, you will be obliged to present evidence that your presence in France is lawful, for which the permesso di soggiorno should suffice, but technically, it seems, that a passport is nonetheless required.

I suspect that your explanation about your passport being in the Irish embassy would normally be accepted by the police, but I also suspect that this may vary according to where you encounter the police, your appearance, the mood of the officer, and so on. If you are very averse to risk, you might want to postpone the trip.

You might also look into whether the Irish embassy will let you have the passport for the trip, after which you can return it to them for the insertion of your visa (or for whatever reason it is that they have it).

No, a residence card is not considered a valid travel document to cross internal EU borders (Artical 21 Convention implementing the Schengen Agreement) .

A residence card, togeather with a valid travel document (passport), shows you are legally in the Schengen Area.

The corresponding French site that states that issued residence permits are accepted, lists only french residence permits as acceptable documents.

The Schengen acquis - Convention implementing the Schengen Agreement of 14 June 1985 between the Governments of the States of the Benelux Economic Union, the Federal Republic of Germany and the French Republic on the gradual abolition of checks at their common borders
...
Article 21

1. Aliens who hold valid residence permits issued by one of the Contracting Parties may, on the basis of that permit and a valid travel document, move freely for up to three months within the territories of the other Contracting Parties, provided that they fulfil the entry conditions referred to in Article 5(1)(a), (c) and (e) and are not on the national list of alerts of the Contracting Party concerned.

Passport and visa requirements
If you are a non-EU national wishing to visit or travel within the EU, you will need a passport:

• valid for at least 3 months after the date you intend to leave the EU country you are visiting,
• which was issued within the previous 10 years,

and possibly a visa.
...
If you have a valid residence permit from one of those Schengen countries, it is equivalent to a visa.

This statement is based on Article 6 - Entry conditions for third-country nationals of the Schengen Border Code.

The Schengen Border Code does not specify anywhere that Article 6 (1)(a)(i) does not apply for residence permit holders visiting other Schengen Countries.

The German residence law AufenthG § 15 (2)(3) quotes Article 6 directly as a reason for Refusal of entry.

Checking the documents of a foreigner: what are the rules?

• (translated to english)

Without any identity check, the police can invite a foreigner of full age to present his papers. He must always have with him the document authorizing his stay in France:

• a passport with a valid visa (unless it is one of the nationalities exempted ) for a stay of 90 days maximum,
• or a residence permit or a long-stay visa (or a receipt or a temporary residence permit) for a stay of more than 3 months.

Note:
The residence permits (further down called title) listed are only those that are issued in France.

The control can only take place on public roads, in public places or open to the public (stations, airports, cafes, etc.).

The police can only carry out a title control directly if objective elements external to the person concerned allow it to be presumed that he is a foreigner. For example, if it drives a vehicle registered abroad or distributes leaflets in a foreign language on the street.

Title controls are limited in time and space. They can not :

• be practiced beyond 6 consecutive hours in the same place,
• lead to the systematic questioning of people present or circulating in this place.

Foreigners who cannot present their residence document can be taken to a police or gendarmerie room and detained there to verify their right to stay.

Sources:

• Convention implementing the Schengen Agreement - EUR-Lex - 42000A0922(02) - EN
• Travel documents for non-EU nationals – Your Europe - Your Europe
• Contrôle des papiers d'un étranger : quelles sont les règles ? | service-public.fr
  • Titles, residence permit and circulation documents for foreigners in France
• AufenthG § 15 Zurückweisung
  • AufenthG Section 15 Refusal of entry
    • the english version states Article 5 instead of Article 6