How does cobra participation affect ObamaCare?

Provided your employer's plan was ACA-compliant (and, as of 1 Jan 2014, it's required to be), if you participate in COBRA after being laid off, it is considered a valid option. Note that you do NOT qualify for any subsidy when using COBRA.

As such, I would expect that overall COBRA participation (i.e. the number of people opting to obtain coverage under the COBRA provisions) will drop significantly except amongst the well-off, as many people will be able to obtain equivalent insurance through the state Exchanges, and either opt for less-expansive (and less expensive) coverage, or qualify to receive a subsidy.

That is, the Exchanges offer virtually all the benefits of the COBRA law. The sole unique benefit of buying via COBRA is that your existing company-supplied plan may not be offered on the exchange, and you'd like to remain on that specific plan. Also, for people who do not qualify for subsidies on the exchanges, it is possible that their costs for insurance on COBRA may be slightly less than the equivalent plan on an Exchange (as the group-level coverage of large employers is still cheaper than that of individual plans from Exchanges).