For what specific travel, lodging and board expenses do workers who were posted or hired out have to be reimbursed?

Principles

Expenses, including the costs of travel, lodging and board, are often incurred by workers from another country (posting country) when posted or hired out to Austria.

Employers are required in certain cases to reimburse workers for costs incurred during periods of posting or hiring-out.
 

Reimbursement of expenses and minimum pay

When workers are posted or hired out, they perform the work agreed and are paid for that work. At the same time, expenses are incurred by them on account of the posting or hiring-out. Workers must be reimbursed for such expenses actually incurred: separately from and in addition to the remuneration they are paid.
Even if reimbursement for expenses is paid out as a lump sum or a general compensation payment, it must not be counted as part of the remuneration for the work performed.

Any allowances specific to posting that are paid out under the laws of the posting country are considered reimbursement for expenses and not part of the remuneration. This applies where the legal basis in the posting country (e.g. a collective agreement) does not make a clear distinction between those items of such an allowance that are paid out to reimburse for expenses actually incurred and those paid out as remuneration.

Are workers entitled to reimbursement for expenses under Austrian law or under the laws of the posting country?

The laws of the country from which workers are posted or hired out (posting country) may well specify reimbursement for expenses actually incurred to the workers when they are posted or hired out.
Where this is governed by provisions in the law or collective agreements in the posting country, or similarly in the employment contract for cases of posting or hiring-out to another country (including Austria), workers are entitled to reimbursement of expenses based on posting country provisions.
This entitlement covers in particular travel expenses from the posting country to Austria, accommodation/lodging expenses in Austria, and return travel from Austria to the posting country.

In addition, Austrian law contains provisions on additional reimbursement of specific expenses (the general requirement being that this must not lead to financial enrichment of the worker concerned):

Under Austrian law, workers posted or hired out are entitled during the entire period of posting or hiring-out to reimbursement for travel, lodging and board expenses incurred as a result of relocating from one place of regular work to another within Austria. Such entitlements to reimbursement of expenses are stipulated in many collective agreements in Austria.
Workers are nonetheless entitled to such reimbursement even if not stipulated in the applicable collective agreement. Where stipulated in the applicable collective agreement, reimbursement of the expenses incurred must be paid out to the workers posted or hired out in the amount stipulated in the relevant Austrian collective agreement.

Workers hired out are additionally entitled to any and all reimbursement of expenses as stipulated in Austrian collective agreements for temporary agency work. For more information on these collective agreements see: Temporary agency work – remuneration.
 

You can search Austrian collective agreements for stipulations on the reimbursement of expenses.