Eloğlu moved from Istanbul to Cologne as a four year old. The name "Şiir" is derived from her father, Metin Eloğlu (1927–1985), a painter and lyricist, who was awarded with important literature awards in the 1960s and 1970s.
After graduating from high school in Cologne, Eloğlu studied for four years at the Hochschule für Musik Saar in Saarbrücken. Her first engagements then took her to theatres in Erlangen, Oberhausen, and Cologne.
In 1991, she made her first film with Kinofilm, Happy Birthday, Türke!, directed by Doris Dörrie and acted with Ulrike Folkerts in her first scene. Eloğlu became known to a wider audience between 1993 and 1996 by her role of young assistant doctor Dr. Nesrin Ergün in the RTL series Stadtklinik.
At the end of 1997, she moved to Berlin. She followed TV and cinema productions as well as theater engagements, among others at the Düsseldorfer Schauspielhaus, where she acted in Antigone and Seven Against Thebes.
Her first international film production was the Swiss film 180° – Wenn deine Welt plötzlich Kopf steht, which was filmed by director Cihan Inan in Zürich. The international cast included Sophie Rois, Christopher Buchholz, Sabine Timoteo, and Michael Neuenschwander. The film ran at festivals in Antalya, Saarbrücken, Sudbury, and Marrakesh, as well as in the Spanish Ourense, where it received an award for Best Screenplay.
She was featured in Almanya - Welcome to Germany, directed by Yasemin Şamdereli, as Leyla Yılmaz. The film was awarded the Deutscher Filmpreis.
From 2014 to 2015, she played the architect Suna Kaya in the soap opera Lindenstraße.
2022
as Ursel
2021
as Farah
2020
as Zehra
2020
as Melek Abu Yazid
2019
as Farah
2018
as Semiha
2017
as Frau Sebnem
2017
2017
as Marie
2017
as Schwiegermutter
2016
as Ayse Nasreddin
2015
as Frau Ülküm
2015
as Hayat
2015
as Nurhan Ecevit
2015
as Bahar Edemir
2014
as Emine Coscun
2014
as Aysun
2014
as Hatice Güngör
2013
as Laila
2013
as Natascha Meisterjahn
2013
as Murat Kurnaz’ Mutter
2012
as Dr. Vogt
2011
as Leyla
2011
as Richterin
2010
as Ülker Özel
2010
as Frau Özlan
2009
as Arzu
2008
as Emine Pascha
2007
as Frau Kelek
2006
as Selime Mölders
2000
as Agent
1997
1993
as Nesrin Ergün
1992
as Ayse
1970
as Frau Mayer
1970
as Irem Gülüc
1970
as Dr. Ipek Aldemir