Palley is a smart guy who I've learned a lot from over the years. But this is not his best work.
By way of preliminaries: MMT consists of three distinct arguments. First is chartalism, the claim that the value of money depends on its acceptability to settle tax obligations. This goes back to G. F. Knapp. Second is functional finance, the claim that government (conceived of as a consolidated budget and monetary authority) seeks to adjust the budget balance to achieve full employment, it can never be prevented from doing so by a financing constraint. This goes back to Abba Lerner and Evsey Domar. And third is the employer of last resort (ELR), a proposal for a specific form of spending to be adjusted under the functional finance rule. This
Personally, I find it useful to set aside the first and third of these arguments and focus on the second, functional finance. My own attempt to restate the functional finance claim in the language of contemporary textbooks is here. In my view, the essential difference between functional finance and orthodoxy is that the assignments of the interest rate and budget instruments are flipped. Instead of setting the interest rate to keep output at potential and setting the budget balance to keep the debt on a sustainable path, we assign the budget balance to keeping output at potential and the interest rate to debt sustainability.
Palley wants to knock down all three planks of MMT. What are his objections to functional finance?
(1) In the absence of economic growth, government deficits will lead to inflation regardless of the output gap. This claim is asserted rather than argued for.  It's not clear what the relevance of this point is, since he agrees that deficits are not inherently inflationary when there is positive growth. Perhaps more importantly, this is a rejection not just as of MMT but of almost all policy-oriented macroeconomics, mainstream and heterodox. Whether you're reading David Romer or Wendy Carlin or Lance Taylor, you're going to find a Phillips curve that relates inflation to current output. There are plenty of disagreements about how expected inflation comes in, but nobody is going to include the budget balance as an independent term. In his eagerness to debunk MMT, Palley here finds himself reasserting Milton Friedman-style monetarism.
(2) If we assume an arbitrary floor on spending and an arbitrary ceiling on taxes, then it may be impossible to achieve both full employment/price stability and a sustainable debt path with interest rates fixed at zero. Yes, this is true. But it proves too much: If we impose arbitrary constraints on tax and spending levels then there is no guarantee that we can achieve debt sustainability and price stability with ANY interest rate. At any given interest rate, there is minimum primary balance that must be achieved to keep output at potential. There is also a minimum primary balance that must be achieved to keep the debt-GDP ratio constant. There is no a priori reason to think the first balance is higher than the second. So Palley's argument here could just as well be a proof that there is nothing government can do to prevent public debt from rising without limit. In any case, these arbitrary limits on taxes and spending are not a feature of standard macro models (including Palley's own models elsewhere), so it's hard to justify bringing them in here.
(3) MMT lacks a theory of inflation. On the contrary, MMT has exactly the same theory of inflation as orthodox macro: High or rising inflation is the result of output above potential, disinflation or deflation the result of output below potential. In other words, MMT is consistent with a standard Phillips curve of the same kind Palley (and almost everybody else) uses. To be fair, the Wray and Tymoigne piece Palley is responding to is not as clear as it might be on this point. But Palley is supposed to be writing a critique of MMT, not just of one particular article. And people like Stephanie Kelton state unambiguously that MMT shares the orthodox output-gap story of inflation; see for instance slide 13 here.
(4) MMT doesn't work in open economies because it requires persistent interest rate differentials between countries. Palley claims that the idea that you can hold interest rates in a given country at zero indefinitely is inconsistent with covered interest parity, a "well-established empirical regularity" that "states there is no room for systematic arbitrage of cross-country interest rates." It seems that Palley has confused covered and uncovered interest parity. CIP is indeed well established empirically, but it only says that there is no arbitrage possible between the spot and forward markets for a given exchange rate. It does not rule out interest-rate arbitrage in the form of the carry trade, and so does not have any implications for the viability of MMT. If UIP held, that would indeed rule out a persistent zero interest policy in the absence of an equally persistent currency appreciation. But UIP, unlike CIP, gets no empirical support in the literature. Japan has sustained the near-zero interest rates that Palley says are unsustainable for 15 years now, and in general, persistent interest rate differentials without any offsetting exchange rate movements are ubiquitous. Furthermore, if financial openness rules out a policy of i=0, then it equally rules out the use of interest rates as a tool for demand management. The best thing you can say about Palley here is that he is parroting orthodoxy; otherwise he is thoroughly confused.
There is one thing that Palley is right about, which is that substantially all of MMT can be found in the old Keynesian literature. This isn't news -- in the same Stephanie Kelton slideshow linked above, she goes out of her way to say that there is nothing "modern" about MMT. And so what? There's nothing wrong with updating insights from the past. In my opinion, most useful work in economics is about pouring old wine in new bottles.
I don't write this from a position within MMT. I tend to feel that the genuine insights of Lernerian functional finance are obscured rather than strengthened by basing them in a chartalist theory of money. It's fine if Tom Palley disagrees with our friends at UMKC and the Levy Institute. But he needs to put down the blunderbuss.
 In fact it's a bit hard to understand what Palley is claiming here. First he says that "money financed deficits" must lead to inflation in a static economy, even with a zero output gap. He adds in a footnote that money-financed are not inflationary in a growing economy; in that case, for price stability "the high-powered money stock must grow at the rate of growth." Then when he writes down a model, this has become the condition that government budget must be balanced, which is something different again. Also, I must say I can't help wrinkling my nose a little, when I read about the "stock of high-powered money," at the smell of a musty antique.
EDIT: Thanks to Daniele Santolamazza for correcting me on the origins of the ELR proposal.