I don't think that the Court's decision today affects whether or not Democrats could filibuster repeal of ObamaCare. Former Senate parliamentarian Robert Dove has discussed the prospect of repealing ObamaCare through reconciliation (which is immune to filibusters): "Anything that reduces the deficit is okay...and nothing that increases the deficit is okay."
Reconciliation is judged on a provision-by-provision basis, not taking into account the legislation as a whole. And it's hard to see how repealing a tax would reduce the deficit. But the deficit would be reduced by repealing the spending provisions of ObamaCare.
On another related matter, the dissent today pointed out that all bills for raising revenue must originate in the House. But they didn't say that this bill originated somewhere other than the House. Weird. I think what happened in Congress was that the House passed some bill, and then the Senate erased the whole thing and substituted a whole new bill; that seems like a constitutionally dubious practice, but maybe none of the plaintiffs complained about it.
Purely as a matter of constitutional law, the Court's recent decision in Miller v. Alabama (regarding life without parole for juvenile murderers) was much more blatantly wrong than the Court's decision today; the Court admitted in that case that the practices it overturned were not "uncommon" (so they couldn't have been cruel and unusual). That doesn't mean I like the decision today, but it will definitely encourage me to work harder to dislodge the incumbent from his incumbency. ObamaCare concentrates too much power in Washington, it's too expensive and it's too inefficient.
UPDATE: For more info about the Origination Clause, see UNITED STATES v. MUNOZ-FLORES, 495 U.S. 385 (1990). I don't think that ObamaCare violated the Origination Clause, given that the original House Bill did impose a tax to fund health care. Perhaps the biggest weak spot in the Chief Justice's opinion today is his characterization of the tax as an "indirect" tax instead of a "direct" tax. After all, a tax on going without health insurance seems equivalent to a tax on personal wealth that was not spent on health insurance. I'm also uncomfortable with the idea that a fine can always be called a "tax" if the fine is small enough. And the Court's example of protective tariffs overlooks that those taxes are closely linked to another enumerated power (i.e. the foreign commerce clause). But the whole issue pretty much seems moot now.
UPDATE #2: I've learned some more about the reconciliation process, and so have added some comments to this blog post, below in the comment thread. Long story short: the GOP might be able to use reconciliation after all, to repeal the individual health insurance tax.