Can I travel within the Schengen area while my Italian residency card is being renewed?

Timatic, the database used by airlines, states the following regarding Italy itself:

Residents of Italy holding expired Residence Permits or a copy of an expired Residence Permit are permitted to re-enter Italy, provided also holding a receipt issued by the Italian Post Office or Police Department, confirming new permit has been applied for. Applications for renewal must be made within 60 days of expiring, otherwise immigration could refuse re-entry, even if holding confirmation of application. Holders copies of an expired Residence Permit also require a statement, stating that they are holding a copy. Residence Permits that include a named child must be valid when traveling with the named child.

This is not stated for Denmark

However, if you travel by air and avoid Ryanair, the chance of being caught by authorities or stopped from flying is next to none - if checking in bags, simply present your passport for them to match you against the ticket, and the same thing if asked to present ID at the gate (only likely with easyJet).

In other words, do not travel by land, as Denmark has re-introduced selective border checks at the land border and ferry ports, and Switzerland (which you would likely pass through) has them as well.