Irish travel document

Visa-free access to the Schengen area by refugees is not harmonised across all Schengen states, and each state still has different requirements.

Of those you mentioned, you can travel visa-free to France, Italy and the Netherlands visa-free for 90 days out of every 180 days. You must travel with both your refugee travel document and IRP (Irish Residence Permit), the red and blue card with a bull and stars above your photograph. Each of those states accepts refugee travel documents issued by Ireland along with the IRP for visa-free entry. Spain, however, does not.

Additionally among Schengen states and candidate states you may travel visa free to Belgium, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Finland, Germany, Greece, Iceland, Liechtenstein, Lithuania, Luxembourg, Malta, Norway, Poland, Portugal, Romania, Slovenia, Sweden, and Switzerland. Some require you to use both your travel document and IRP; others require only the travel document. The details for each are below.

The relevant visa exemptions were found in Timatic, the database which airlines use to verify passenger travel documents. This means that any airline will be aware of the visa exemption and should not deny boarding.

Note that I did not find a visa exemption for Austria, Estonia, Latvia, Slovakia, or Spain. You should not attempt to fly directly to these countries from outside the Schengen area, not even to transit to another Schengen state, and you should consider not travelling to them via ground transportation or internal Schengen flights.

Belgium (BE)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Austria, Bulgaria, Croatia,
  Cyprus, Czechia, Denmark, Estonia, Finland, France, Germany,
  Greece, Hungary, Iceland, Ireland (Rep.), Italy, Latvia,
  Liechtenstein, Lithuania, Luxembourg, Malta, Netherlands,
  Norway, Poland, Portugal, Romania, Slovakia, Slovenia,
  Spain, Sweden, Switzerland or the United Kingdom, for a
  maximum stay of 90 days.(SEE NOTE 56075)
  NOTE 56075: The maximum stay is granted within 180 days.

Bulgaria (BG)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or an EEA Member
  State for a maximum stay of 90 days. (SEE NOTE
  56110)
  NOTE 56110: The maximum stay is granted within 180 days.

Croatia (HR)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Andorra, Monaco, San Marino,
  Switzerland, Vatican City (Holy See) and another EEA Member
  State for a maximum stay of 90 days. (SEE NOTE
  53527)
  NOTE 53527: The maximum stay is granted within 180 days.

Cyprus (CY)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Andorra, Antigua and Barbuda,
  Argentina, Australia, Bahamas, Barbados, Brazil, Brunei
  Darussalam, Canada, Chile, Costa Rica, Dominica, El
  Salvador, Grenada, Guatemala, Honduras, Israel, Japan, Korea
  (Rep.), Malaysia, Mauritius, Mexico, Monaco, New Zealand,
  Nicaragua, Panama, Paraguay, Samoa, San Marino, Seychelles,
  Singapore, St. Kitts and Nevis, St. Lucia, St. Vincent and
  the Grenadines, Switzerland, Timor-Leste, Trinidad and
  Tobago, United Arab Emirates, Uruguay, Vanuatu, Vatican City
  (Holy See) and Venezuela or another EEA Member State
  for a maximum stay of 90 days. (SEE NOTE 66301)
  NOTE 66301: The maximum stay is granted within 180 days.

Czechia (CZ)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Bulgaria, Croatia, Cyprus,
  Ireland (Rep.), Romania or another Schengen Member State
  for a maximum stay of 90 days. (SEE NOTE
  53652)
  NOTE 53652: The maximum stay is granted within 180 days.

Denmark (DK)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees and a residence permit issued by
  another EEA Member State, Switzerland or the
  United Kingdom, for a maximum stay of 90 days. (SEE NOTE
  56210) (SEE NOTE 53755)
  NOTE 53755: This does not apply when traveling to the
  Faroe Isl. or Greenland.
  NOTE 56210: The maximum stay is granted within 180 days.

Finland (FI)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 56245)
  NOTE 56245: The maximum stay is granted within 180 days.

France (FR)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees and a residence permit issued by
  Andorra, Monaco, Switzerland or EEA Member State
  for a maximum stay of 90 days. (SEE NOTE
  53781)
  NOTE 53781: The maximum stay is granted within 180 days.

Germany (DE)

• Passengers with a Travel Document (Convention of 28 July
  1951), valid for at least 4 months from the arrival date,
  issued to refugees by Andorra, Antigua and Barbuda,
  Argentina, Australia, Bahamas, Barbados, Brazil, Brunei
  Darussalam, Canada, Chile, Costa Rica, El Salvador,
  Guatemala, Honduras, Hong Kong (SAR China), Israel, Japan,
  Korea (Rep.), Macao (SAR China), Malaysia, Mauritius,
  Mexico, Monaco, New Zealand, Nicaragua, Panama, Paraguay,
  San Marino, Seychelles, Singapore, St. Kitts and Nevis,
  Switzerland, Uruguay, Vatican City (Holy See), Venezuela or
  another EEA Member State for a maximum stay of
  90 days. (SEE NOTE 53546)
  NOTE 53546: The maximum stay is granted within 180 days.

Greece (GR)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 56269)
  NOTE 56269: The maximum stay is granted within 180 days.

Iceland (IS)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or an EEA Member
  State for a maximum stay of 90 days. (SEE NOTE
  56309)
  NOTE 56309: The maximum stay is granted within 180 days.

Italy (IT)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees and a residence permit issued by
  Switzerland or another EEA Member State (excluding the United Kingdom) for a maximum stay of 90 days. (SEE
  NOTE 56322)
  NOTE 56322: The maximum stay is granted within 180 days.

Liechtenstein (LI)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Ireland (Rep.), Romania or
  United Kingdom.

Lithuania (LT)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 53555)
  NOTE 53555: The maximum stay is granted within 180 days.

Luxembourg (LU)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 56336)
  NOTE 56336: The maximum stay is granted within 180 days.

Malta (MT)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 56340)
  NOTE 56340: The maximum stay is granted within 180 days.

Netherlands (NL)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees and a residence permit issued by
  Switzerland or another EEA Member State for a
  maximum stay of 90 days. (SEE NOTE 56347)
  NOTE 56347: The maximum stay is granted within 180 days.

Norway (NO)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 56368)
  NOTE 56368: The maximum stay is granted within 180 days.

Poland (PL)

• Passengers with a biometric Travel Document (Convention of
  28 July 1951) issued to refugees by Ireland (Rep.) for a
  maximum stay of 90 days. (SEE NOTE 52786)
  NOTE 52786: The max. stay is granted within 180 days.

Portugal (PT)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Switzerland or another EEA
  Member State for a maximum stay of 90 days.
  (SEE NOTE 51200)
  NOTE 51200: The maximum stay is granted within 180 days.

Romania (RO)

• Passengers with a travel document issued to refugees or
  stateless persons by Ireland (Rep.) or United Kingdom for a
  maximum stay of 90 days. (SEE NOTE 53643)
  NOTE 53643: The maximum stay is granted within 180 days.

Slovenia (SI)

• Passengers with a travel document issued to refugees or
  stateless persons by Switzerland or another EEA Member State
  for a maximum stay of 90 days. (SEE NOTE 53647)
  NOTE 53647: The maximum stay is granted within 180 days.

Sweden (SE)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Austria, Belgium, Czechia,
  Denmark, Estonia, Finland, France, Germany, Hungary, Ireland
  (Rep.), Iceland, Italy, Latvia, Liechtenstein, Lithuania,
  Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,
  Romania, Slovenia, Slovakia, Spain and Switzerland for a
  maximum stay of 90 days. (SEE NOTE 56388)
  NOTE 56388: The maximum stay is granted within 180 days.

Switzerland (CH)

• Passengers with a Travel Document (Convention of 28 July
  1951) issued to refugees by Ireland (Rep.), Romania or
  United Kingdom.