The effect on the efficacy and potency of a non-competetive antagonist binding to the active site of the receptor (dose-response curve)
This is a tough question. I was reading this paper Patrono, C., et al. "Clinical pharmacology of platelet cyclooxygenase inhibition." Circulation 72.6 (1985): 1177-1184. and they seemed t mention this paragraph in the introduction.


  Platelet Cycloxygenase or prostaglandin (PG) H synthase (i.e., the
  enzyme that converts arachidonate released from membrane phospholipids
  into PG endoperoxides) is the target of several antiplatelet agents
  that can reversibly or irreversibly block the activity of the enzyme
  by competing with the substrate or permanently altering the active
  site, respectively.Such drugs belong to the class of so-called
  nonsteroidal anti-inflammatory agents (e.g., indomethacin, aspirin),
  but also include a uricosuric agent, i.e., sulfinpyrazone.


So we have to look for  DRC of a NSAID to say COX. Now I was unable to find a good DRC for aspirin (If you find it, let me know in the comments below. I would love to take a look at it). But I did find DRC for other drugs that inhibit COX.

It led me to this paper Walker, M., et al. "A three-step kinetic mechanism for selective inhibition of cyclo-oxygenase-2 by diarylheterocyclic inhibitors." Biochem. J 357 (2001): 709-718. which described the inhibiton of COX2 by a NSAID called celecoxib.


  The inhibition of COX isoenzymes by NSAIDs generally conforms to one of three inhibitory mechanisms: simple reversible inhibition, as demonstrated by ibuprofen ; time-dependent reversible inhibition, which includes both weaker binding inhibitors, such as naproxen , and tight binding inhibitors, such as indomethacin and meclofenamic acid ; and irreversible covalent inhibition, as demonstrated by aspirin  and o-(acetoxy-phenyl)hept-2-ynyl sulphide ('APHS') . Most of the traditional NSAIDs display similar inhibitory mechanisms against both COX-1 and COX-2 , and are relatively non-selective. However, COX-2-selective diarylheterocyclic inhibitors demonstrate distinct inhibitory mechanisms for the two isoforms . For example, celecoxib  has been reported to be a reversible competitive inhibitor of COX-1 while demonstrating time-dependent irreversible inhibition of COX-2. 


Considering the highlighted statement, I think you can consider the COX1 curve as a standard competitive inhibition and COX2 as non competitive inhibition. The resulting change is essentially the curve shape itself (being shifted to the right). 

The DRCs were reported here: Tacconelli, Stefania, et al. "The biochemical selectivity of novel COX-2 inhibitors in whole blood assays of COX-isozyme activity." Current Medical Research and Opinion® 18.8 (2002): 503-511.



Interestingly the second paper (By Walker) also described how (the authors proposed) the binding occurs. A very interesting read.