Simply state that you are a NEMO European Museum Conference participant and show your badge for free entry at the Mo Museum and the several branches of the Vilnius City Museum and the National Museum of Lithuania:

• MO museum
  Private museum of modern and contemporary Lithuanian art in Vilnius, home to one of the country's largest art collections and offering exhibitions, events, films, and educational programmes in an accessible and interactive setting.
  Opening hours: Daily 10:00–20:00 (Except Tuesday closed)
  Address: Pylimo g. 17, 01141 Vilnius
  Visit the website

Vilnius City Museum and its three branches:

• Vilnius City Museum (Vokiečių 6)
  Hosts two or three exhibitions about Vilnius and its residents every year, telling the story of the city and its people. Vokiečių 6 invites visitors to look into their everyday surroundings and find something unexpected.
  Opening hours: Saturday-Sunday 11:00-19:00, Tuesday-Friday: 15:00-19:00
  Address: Vokiečių st. 6
  Visit the website
• Center for Wooden Architecture
  Presents, documents and fosters the tradition of wooden architecture in Vilnius, and promotes contemporary wooden architecture in the city.  
  Opening hours: Saturday-Sunday 11:00-19:00, Tuesday-Friday: 15:00-19:00
  Address: 52 Polocko St.
  Visit the website
• Markučiai Manor
  An authentic example of 19th-century wooden architecture nestling in its own park. The house has an exhibition focusing on the district of Markučiai and the story of an engineer’s family in the Russian Empire.
  Opening hours: Saturday-Sunday 11:00-20:00, Tuesday-Friday: 15:00-20:00 (English version of the website says it's closed, but it is open). 
  Address: 124 Subačiaus St.
  Visit the website
• Beatrice’s House
  A typical three-room flat, once the home of the partially sighted singer Beatričė Grincevičiūtė (1911–1988). The exhibition paints a picture of cultural life in Soviet Vilnius, the housing policies of the time, and the challenges faced by those with a visual impairment.
  Opening hours: Saturday-Sunday 11:00-19:00, Tuesday-Friday: 15:00-19:00
  Address: 12 Vienuolio St
  Visit the website

The 10 branches of the National Museum of Lithuania:

• Castellan’s House
  A short and interesting guide to Lithuania, introducing the main ideas that have shaped Lithuania’s identity. We invite you to discover the face of Lithuania.
  Opening hours: Tuesday-Sunday 10:00-18:00
  Address: Arsenalo g. 1
  Visit the website
• Gediminas Castle Tower
  Gediminas Castle Tower is the most frequently visited branch of the National Museum of Lithuania. Its exhibition invites visitors to learn about the history of Vilnius as the centre of the Grand Duchy of Lithuania, and also to take in the most spectacular panoramic view of the city.
  Opening hours: Saturday-Sunday 11:00-19:00, Tuesday-Friday: 15:00-19:00
  Address: Everyday 10:00-18:00
  Visit the website
• Old Arsenal
  A 12,000–year journey through prehistoric Lithuania
  Opening hours: Tuesday-Sunday 10:00-18:00
  Address: Arsenalo g. 3
  Visit the website
• House of Histories
  The House of Histories is a space which invites visitors to experience history not as something fixed in the past but as integral to the reality we are creating today.
  Opening hours: Tuesday-Sunday 10:00-18:00 (Wednesdays: 10:00-20:00)
  Address: T. Kosciuškos g. 3
  Visit the website
• House of Signatories
  A short and interesting guide to Lithuania, introducing the main ideas that have shaped Lithuania’s identity. We invite you to discover the face of Lithuania. Although the museum will be temporarily closed while a new permanent exhibition is being installed, the House of Signatories Gallery is open.
  Opening hours: Tuesday-Friday 10:00-18:00, Saturday-Sunday 12:00-17:00
  Address: Pilies g. 26
  Visit the website
• Bastion of the Vilnius Defence Wall
  The museum at the Bastion of the Vilnius Defence Wall tells the history of the Grand Duchy of Lithuania’s weaponry and defence practices.
  Opening hours: Tuesday-Sunday 10:00-18:00
  Address: Bokšto g. 20
  Visit the website
• Kazys Varnelis House-Museum
  At first glance, the house–museum of the Lithuanian émigré artist and collector Kazys Varnelis looks like a small residential building. In fact, though, it contains more than 40 exhibition spaces combining antique graphic arts, maps, Western European sculpture and painting, historic furniture and Asian artworks, as well as Varnelis’s own optical art compositions.
  Opening hours: Tuesday-Sunday 10:00-18:00
  Address: Didžioji g. 26
  Visit the website
• Jonas Basanavičius Birthplace
  Outside of Vilnius: Located in the village of Ožkabaliai, Vilkaviškis region, the birthplace of Jonas Basanavičius is one of only three National Museum of Lithuania branches located outside of Lithuania’s capital.
  Opening hours: Tuesday-Friday 10:00-17:00, Saturday 10:00-15:00
  Address: Gimtinės g. 17, Ožkabalių I k., Vilkaviškio r.
  Visit the website
• Vincas Kudirka Museum
  Outside of Vilnius: The Vincas Kudirka Museum introduces visitors to Vincas Kudirka, the author of the Lithuanian anthem, as well as the town in which he spent the final years of his life.
  Opening hours: Tuesday-Friday 10:00-17:00, Saturday 10:00-15:00
  Address: V. Kudirkos g. 29, Kudirkos Naumiestis, Šakių r.
  Visit the website
• Jonas Šliūpas Museum
  Outside of Vilnius: Visit the 19th century wooden villa built by counts Tiškevičiai, which was the home of Palanga’s first burgomaster Jonas Šliūpas from 1931 to 1944, and feel the spirit of this progressive and uncompromising personality. The permanent exhibition about Jonas Šliūpas is currently in development. Meanwhile, visitors are invited to visit the museum’s temporary exhibitions.
  Opening hours: Wednesday-Friday 11:00-17:00, Saturday-Sunday 11:00-16:00
  Address: Vytauto g. 23a, Palanga
  Visit the website