What are some examples of nations that involuntarily gained independence?

Canada!

Little bit further out there, but I'll refer to this link here: Canadian Independence

Canada celebrate it's conception (canada day) as 1867, however it was still at the whims of the British parliament. Actual independence would come about somewhere between 1919 and 1931:

Canada's transition from a self-governing British colony into a fully independent state was an evolutionary process, which arose in such a gradual fashion that it is impossible to ascribe independence to a particular date. The Supreme Court of Canada reflected this uncertainty when it said in Re Offshore Mineral Rights of British Columbia that Canada's "sovereignty was acquired in the period between its separate signature of the Treaty of Versailles in 1919 and the Statute of Westminster, 1931..."[1] However, the development of this independence had its roots before 1919, and was not actually completed until well after 1931. As Frank Scott has argued, "Never at any time in [1919-39] was the full international personality of the Dominions, as distinct from Great Britain, established beyond equivocation"

It's a bit odd that the supreme court of Canada has to give a date range on it's independence from Britain. And even then, British law could still be passed that would apply to Canada

Indeed, symbolically-important legal traces of Canada's colonial status were only shed with the passing of the Canada Act[3] by the British Parliament in 1982. That Act not only provided for the first time a process by which Canada's basic constitutional laws could be legally amended without action by the British Parliament, but it also declared that no British law passed thereafter would apply to Canada. There are still two final vestiges of colonialism to be eliminated, those found in ss.55 and 56 of the 1867 Constitution Act which provide for the reservation and disallowance of federal legislation. Of course Canada has been an independent nation for a number of decades, and these shadows of her former status are nothing more than anomalies which illustrate how the legal provisions of the Canadian constitution failed to keep pace with the political developments which propelled Canada to full statehood.

I'm not putting forward that Canada did not want it's independence, however as far as independent nations go, Canada sure took it's sweet time in splitting off from the British (115 year process?). Still to this day, the Governor General represents the Monarch in Canada (largely symbolic, yet still involved). Public sentiment has only just shifted towards severing these ties (admittedly only after the current queen passes).

Does clinging on to the British parliament and royalty for as long as they have count as involuntary? I'll let +1 - 1 decide.

There is some difficulty in how the term nation is defined and used here.

Mainly adding to Alex' answer:

It might be argued that Austria was an entity that was in modern times forced out of its home nation (whatever 'Germany' was at that time) three times additionally to what Alex identified!

Austria has been a member of the Holy Roman Empire from its inception.

1. It was forced out of this Reich first by Napoleon in 1806
2. Then a German Confederation was established as a successor with Austria as a leading power. But as that construct slowly developed into the German Reich Austria was again forced out by the by Bismarck in ~1866–71,
3. When Austria-Hungary itself collapsed in 1918 the allies after World War I in 1919 forced Austria out of Germany again:

On November 13, 1918, German-Austria asked Germany to start negotiations of union and on November 15 sent a telegram to President Wilson to support union of Germany and Austria. This was grounded in the view that Austria had never been a nation in the true sense. While the Austrian state had existed in one form or another for over 700 years (dating to the Holy Roman Empire), its only unifying force had been the Habsburgs. Apart from being German-inhabited, these Lands had no common "Austrian" identity. They were Habsburg-ruled lands that had not joined the Prussian-dominated German Empire after the Austrian Empire lost the Austro-Prussian War.

On March 12, 1919, the Constituent Assembly re-confirmed an earlier declaration that German-Austria was a constituent part of the German republic. Pan-Germans and Social Democrats supported the union with Germany, while Christian Socialists were less supportive.

During spring and summer of 1919, unity talk meetings between German and Austrian representatives continued. All this changed after June 2, 1919 when the draft peace treaty with Austria was presented, which demonstrated that the Western Allies were opposed to any union between Germany and Austria.
_{–– Wikipedia: Republic of German-Austria # Failed union with Germany}

And again in addition to Alex: Serbia was the dominant state of Yugoslavia and tried very hard to not only remain in this state, but also to be that state.

As the Yugoslav Wars raged through Croatia and Bosnia, the republics of Serbia and Montenegro, which remained relatively untouched by the war, formed a rump state known as the Federal Republic of Yugoslavia (FRY) in 1992. The Federal Republic of Yugoslavia aspired to be a sole legal successor to the Socialist Federal Republic of Yugoslavia, but those claims were opposed by the other former republics.

The United Nations also denied its request to automatically continue the membership of the former state. In 2000, Milosevic was prosecuted for atrocities committed in his ten-year rule in Serbia and the Yugoslavia War. Eventually, after the overthrow of Slobodan Milošević from power as president of the federation in 2000, the country dropped those aspirations, accepted the opinion of the Badinter Arbitration Committee about shared succession, and reapplied for and gained UN membership on 2 November 2000. (From 1992 to 2000, some countries, including the United States, had referred to the FRY as Serbia and Montenegro.) In April 2001, the five successor states extant at the time drafted an Agreement on Succession Issues, signing the agreement in June 2001. Marking an important transition in its history, the Federal Republic of Yugoslavia was officially renamed Serbia and Montenegro in 2003.
_{–– Wikipedia: Yugoslavia # Breakup}

The German states in 1806. They were made independent involuntarily when the holy Roman Empire was dissolved. Germans and treated that news like it was a national tragedy and the end of Germany as a country. The emperor made the decision to do it unilaterally because of Napoleon's victories without consulting the princes. The Germans were so upset that some of them refused to recognize the dissolution.

https://en.wikipedia.org/wiki/Dissolution_of_the_Holy_Roman_Empire

However the German states were already very independent so very little actually changed in reality.

Austria in 1918 obtained independence from the Austro-Hungarian empire, as a result of lost war.

Also Russia in 1992 became independent from Soviet Union, which collapsed. It was clearly against the will of most Russians.

There is only one such case in history according to my research.

Singapore is arguably the only modern nation to have been forced against its will into independence: The One Country That Gained Independence against Its Will.

The island became independent on 9 August 1965, thus becoming the only country to gain independence against its own will in the history of the modern world! - Singapore (新加坡).

Slovakia in 1992. Well, kind of. The (perhaps) only party that had independence on its political agenda was the Slovak National Party (8% votes), the outright independence was neither supported nor expected by any majority of population (polls gave anything between 10 to 30% in support, depending on the question and time period), and it was basically a decision of the Czech part (maybe even Václav Klaus personally) when they realized HZDS/Mečiar is blocking and obstructing everything and the negotiations about the future of the federation lead to nowhere.

Bantustans were carved out of South Africa by the insistence of South African apartheid government, and some (4) of them granted independence. It is difficult to comment on the willingness of the Bantustans themselves, given the circumstances.

Kazakhstan could be considered one such country.

While the dissolution of the USSR was a fairly long process that included many republics declaring themselves sovereign within the Union, the secessions started in 1990 with Lithuania, Estonia, and Latvia jumping ship.

In August 1991, the Soviet government made one last attempt to regain power by staging a coup against Gorbachev, but it failed, and with it, the Communist Party effectively ceased to function. The remaining republics quickly deserted. On December 8th, when the Belavezha Accords were signed, only two republics had yet to declare independence - Russia and Kazakhstan. Russia would officially declare its independence from the USSR four days later, with the parliament ratifying the accords.

This left Kazakhstan as the only member republic from December 12th to December 16th, when it also declared independence from the USSR. Though the USSR did not officially vanish until December 26, it could be argued that it stopped being a union as soon as there was only one member, and Kazakhstan was independent as soon as Russia checked out.

The degree to which Panama sought its own independence from Colombia / Gran Colombia may pale next to the interests of the U.S. in separating it. In other words, Panamanian independence from Colombia was largely a U.S. project to protect its Canal interests.

Britain (sort of)

In 410 CE, the Roman emperor Honorius refused to send soldiers to the island to defend it, basically leaving it to fend for itself against the invaders from across the sea. This effectively led to the end of Britannia as a Roman province.