For the theory it's based upon - very, very little.
For the act of "talking therapy" itself there is some evidence of benefit from studies and meta analysis to suggest it helps but this in itself does not indidcate that psychodynamic theory is valid.
In short it's nonsense. But it is nonsense that in some cases can work.
ETA: Apologies for lack of sources, it's just these things seem so remarkably trivial as I am so familiar with them.
For instance see: The efficacy of psychodynamic psychotherapy.
Shedler, Jonathan American Psychologist, Vol 65(2), Feb-Mar 2010, 98-109. http://www.apa.org/pubs/journals/releases/amp-65-2-shedler.pdf
Abstract:
Empirical evidence supports the efficacy of psychodynamic therapy. Effect sizes for >psychodynamic therapy are as large as those reported for other therapies that have been >actively promoted as “empirically supported” and “evidence based.” In addition, patients >who receive psychodynamic therapy maintain therapeutic gains and appear to continue to >improve after treatment ends. Finally, nonpsychodynamic therapies may be effective in part >because the more skilled practitioners utilize techniques that have long been central to >psychodynamic theory and practice. The perception that psychodynamic approaches lack >empirical support does not accord with available scientific evidence and may reflect >selective dissemination of research findings. (PsycINFO Database Record (c) 2010 APA, all >rights reserved)
and in comparison with pharmacological interventions: Relative efficacy of psychotherapy and pharmacotherapy in the treatment of depression: A meta-analysis
Psychotherapy Research
Volume 16, Issue 5, 2006, Pages 566 - 578
Authors: Saskia De Maata; Jack Dekkera; Robert Schoeversa; Frans De Jonghea
DOI: 10.1080/10503300600756402 http://www.informaworld.com/smpp/content~db=all?content=10.1080/10503300600756402
Abstract:
We investigated the efficacy of pharmacotherapy and psychotherapy for depression by >searching for RCT's. Studies were classified according to chronicity and severity and a >meta-analysis was applied. Ten studies were included. Remission did not differ between >psychotherapy (38%) and pharmacotherapy (35%). No differences were found in chronic, or in >non-chronic depression, and in mild or in moderate depression. Both treatments performed >better in mild than in moderate depression. Dropout was larger in pharmacotherapy (28%) >than in psychotherapy (24%). At follow-up relapse in pharmacotherapy (57%) was higher than >in psychotherapy (27%). Psychotherapy and pharmacotherapy appear equally efficacious in >depression. Both treatments have larger effects in mild than in moderate depression, but >similar effects in chronic and non-chronic depression and at follow-up psychotherapy >outperforms pharmacotherapy.
As for the assertion that theoretically it's nonsense:
Efficacy of treatment depends on therapist not school of psychoanalysis The alliance.
Horvath, Adam O. Current issue feed Psychotherapy: Theory, Research, Practice, Training, Vol 38(4), Win 2001, 365-372. http://psycnet.apa.org/?&fa=main.doiLanding&doi=10.1037/0033-3204.38.4.365
Also psychoanalytic therapies have, by and large - anorexia is a notable exception, been abandoned by psychologists in favour of CBT and mindfulness treatments.
Some of Freud, and other early psychoanalysts, theories influenced the direction of psychology and you can see some vestiges of psychoanlytic ideas in some mainstream psychological theories. But I would say vestiges only and that psychology, for the main has abandoned psychodynamic theory as valid or having any explanatory power.
That it was built on unfalsifiable case studies doesn't help it's cause either