Effect would be that Polk still is the Democratic candidate in 1844 and wins the election against Harrison.
Harrison would not be a candidate for re-election in 1844--he was quite emphatic about that. Read his Inaugural Address:
"I proceed to state in as summary a manner as I can my opinion of the sources of the evils which have been so extensively complained of and the correctives which may be applied. Some of the former are unquestionably to be found in the defects of the Constitution; others, in my judgment, are attributable to a misconstruction of some of its provisions. Of the former is the eligibility of the same individual to a second term of the Presidency. The sagacious mind of Mr. Jefferson early saw and lamented this error, and attempts have been made, hitherto without success, to apply the amendatory power of the States to its correction. As, however, one mode of correction is in the power of every President, and consequently in mine, it would be useless, and perhaps invidious, to enumerate the evils of which, in the opinion of many of our fellow-citizens, this error of the sages who framed the Constitution may have been the source and the bitter fruits which we are still to gather from it if it continues to disfigure our system. It may be observed, however, as a general remark, that republics can commit no greater error than to adopt or continue any feature in their systems of government which may be calculated to create or increase the lover of power in the bosoms of those to whom necessity obliges them to commit the management of their affairs; and surely nothing is more likely to produce such a state of mind than the long continuance of an office of high trust. Nothing can be more corrupting, nothing more destructive of all those noble feelings which belong to the character of a devoted republican patriot. When this corrupting passion once takes possession of the human mind, like the love of gold it becomes insatiable. It is the never-dying worm in his bosom, grows with his growth and strengthens with the declining years of its victim. If this is true, it is the part of wisdom for a republic to limit the service of that officer at least to whom she has intrusted the management of her foreign relations, the execution of her laws, and the command of her armies and navies to a period so short as to prevent his forgetting that he is the accountable agent, not the principal; the servant, not the master. Until an amendment of the Constitution can be effected public opinion may secure the desired object. I give my aid to it by renewing the pledge heretofore given that under no circumstances will I consent to serve a second term."
http://www.bartleby.com/124/pres26.html
(I assume that Clay will be the Whig candidate in 1844. Webster is a potential competitor, but his background as a Federalist would hurt him--it would play into the hands of the Democrats who argued that the Whigs were just Federalists under a new name.)
It is also rather unlikely that Polk would be the Democratic candidate in 1844. In OTL, he was nominated largely because of dissatisfaction in the South over Van Buren's hedging over Texas. Texas would never have become such a big issue with southerners if President Tyler and his Secretary of State Calhoun had not used the "bully pulpits" of the presidency and the Secreatry of State's office to scare them about British plots to abolish slavery there...
(Admittedly, there were other reasons for dissatisfaction with Van Buren. The Cass supporters disliked his hard-money policies; and many Democrats worried about his unpopularity due to the hard times during his administration. Still, without Texas I think he would have won the nomination.)