FC Barcelona is the European football team that has the longest consecutive participation streak of 60 (1955/58-2017/18) and counting in UEFA club competitions.
Data from the linked websiteRec.Sport.Soccer Statistics Foundation upto season 2015/16:
Consecutive Participations
58 Barcelona (1955/58-2015/16)
52 Anderlecht (1964/65-2015/16)
42 FC Porto (1974/75-2015/16) PSV (Eindhoven) (1974/75-2015/16)
41 Benfica (1960/61-2000/01)
36 Sporting (Lisboa) (1977/78-2012/13)
33 Sparta Praha (1983/84-2015/16)
31 Rangers (1981/82-2011/12)
28 Juventus (1963/64-1990/91)
27 Dynamo Kyiv (1989/90-2015/16)
25 Ajax (1991/92-2015/16)
24 Ajax (1966/67-1989/90) [banned from 1990/91
competition] Crvena zvezda Beograd (1968/69-1991/92) [banned
1992/93 through UN boycott] Spartak Moskva (1980/81-2003/04)
Manchester United (1990/91-2013/14) Olympiakos (Piraeus)
(1992/93-2015/16)
23 Levski Sofia (1991/92-2013/14) Bayern München (1993/94-2015/16)
22 Real Madrid (1955/56-1976/77) FC Brugge (1994/95-2015/16)
21 Liverpool (1964/65-1984/85) [banned 1985/86-1991/92
(would have qualified each year)] AEK (Athinai) (1991/92-2011/12)
Celtic (1995/96-2015/16)
20 Austria (Wien) (1976/77-1995/96) Flora Tallinn (1994/95-2013/14)
Arsenal (1996/97-2015/16) Partizan Beograd (1996/97-2015/16)
19 Crvena zvezda Beograd (1995/96-2013/14) [banned from 2014/15 competition] Chelsea (1997/98-2015/16) Olympique Lyonnais
(1997/98-2015/16) Real Madrid (1997/98-2015/16) Shakhtar Donetsk
(1997/98-2015/16)
18 Malmö FF (1964/65-1981/82) Real Madrid (1978/79-1995/96)
Slavia Praha (1992/93-2009/10)
17 Omonia Nicosia (1975/76-1991/92) [withdrew from 1974/75
competition] Besiktas (1984/85-2000/01) Rosenborg BK (Trondheim)
(1989/90-2005/06) F91 Dudelange (1999/00-2015/16) Sheriff
Tiraspol (1999/00-2015/16)
16 Celtic (1962/63-1977/78) CSKA (Sofia) (1969/70-1984/85)
Internazionale (1976/77-1991/92) Steaua Bucuresti (1984/85-1999/00)
Feyenoord (1991/92-2006/07) Skonto Riga (1992/93-2007/08) The
New Saints (2000/01-2015/16) FK Ventspils (2000/01-2015/16)
15 Sporting (Lisboa) (1961/62-1975/76) Grasshopper
(1970/71-1984/85) Aberdeen (1977/78-1991/92) Dinamo Bucuresti
(1981/82-1995/96) Panathinaikos (1998/99-2012/13) Milan
(1999/00-2013/14) FC København (2001/02-2015/16)