Hitler renounced his Austrian citizenship in 1925, noting that he never felt Austrian (which is one reason why he enlisted in the Bavarian Army during the First World War and not the Austrian). He was then stateless until 1932, as he never took up German citizenship. This of course meant he couldn't actually run for office in this time, and his citizenship was an issue during his trail after the Beer Hall Putsch in 1924 (he could have been deported, being a foreigner and all). However he decided to run for President in 1932 against Paul von Hindenburg, and to do so he was required to obtain German citizenship, which he did via a nominal position as a government councillor in Brunswick and a state representative in Berlin.

Reading Kershaw's biography, he simply notes that Hitler had previously considered gaining German citizenship in both 1929 and 1930, but doesn't explain why he didn't follow through, or if there was some issue in doing so. Nor does he note any problems arising from Hitler's statelessness in this time, just that he couldn't pass up a chance to run for the presidency in 1932, even if it meant going against an ardent nationalist like Hindenburg. Of course I'm citing the condensed one-volume version of Kershaw's biography, so the two-volume version may have more detail (Kershaw notes he cut some 650 pages of the 1450 total to make this edition), but I suspect that's unlikely.

Ian Kershaw. Hitler, 2008.

There's an important thing the other thread didn't touch on - an Austrian identity separate from a German identity didn't gain much traction until after the second world war. (In Austria-Hungary, they had their monarch, but so did many other nationalities, it wasn't very different from being Bavarian or Prussian, just subsets of Germanness...) and the other German states were only unified because of the Franco Prussian war of 1871, I would suggest you look up "Großdeutschland" and "Kleindeutschland" to understand why Austria didn't join also.

Think of people fighting for Texan independence, establishing a Texan state and so on, they then joined the USA. Hitler as an Austrian was as a Texan citizen at the time, this is why the Anschluss eventually happened - it had been the goal of German nationalists for 100 years. At the end of WWI, the constituent assembly e.g. of the new state of "Deutschösterreich" (German-Austria) attempted to unify with Germany, which the Entente powers put a stop to it, in both the Treaty of St. Germain and Versailles.

By, "Was this ever a problem for him?" do you mean, "Did this conflict with Nazi ideology about German nationalism?" Then no, not really. Central to Nazi ideology was the concept of a Greater Germany, which was an ideal nation-state which would unify all German-speaking peoples into one powerful nation. Prior to Bismarck's chancellorship in Prussia, Germany had been a loose confederation of small states, with Prussia and Austria being the two main exceptions. The Prussian Empire under Bismarck used a growing tide of German nationalism to its advantage in order to grow into what we now call Germany by annexing smaller German states over time, though before this Austria was generally viewed as a part of the greater German region. Indeed, many German nationalists during the period of unification had aspired to annex Austria as well, but were unable to since it was then the center of the Hapsburg Austro-Hungarian Empire, which would not be split up until after World War I. In the view of Hitler and his followers, Austria was not a foreign nation but rather a rightful part of the greater German nation, hence the Nazi movement for unification which would culminate in the Anschluss in 1938. Hitler's Austrian origin was therefore not viewed by him or his supporters as being in conflict with his German identity.

Mark Mazower, Hitler's Empire (New York: Penguin, 2008)