Bottom line: Tourists, visitors, and transit cases are not affected by the UK's vote to leave the European Union.

The relevant issues are about asylum seekers and some types of EU nationals who seek to gain (or persist) settlement in the UK. If you are an EU national exercising derived rights or exercising treaty rights, please use Expats for your questions/answers.

Update 19 August 2016

The Home Affairs Committee published a report dated 27 July 2016, The work of the Immigration Directorates (Q1 2016), which concludes there is an "absence of certainty" over Brexit. It mainly deals with the status of EU nationals who are now exercising treaty rights in the UK.

Update 2 July 2016

What a lot of people are interested in now has a name: "domestic disentanglement from EU law". The House of Lords has added a briefing to their library...

Repealing and Reviewing Domestic Legislation—As part of the process of leaving the EU, decisions would need to be made about how to deal with existing domestic legislation passed to enable EU law to have effect in the UK, a process which the House of Lords European Union Committee has described as "domestic disentanglement from EU law". Parliament would have an important role to play in reviewing, repealing, amending and replacing legislation, a process which is predicted by many to be complex and time-consuming. Once the UK had formally triggered Article 50, its timescales would apply independently of Parliament approving domestic legislative changes associated with leaving the EU.

The full briefing is here.

Update 28 June 2016

The House of Commons has admitted a research paper to their library: "Leaving the EU: How might people currently exercising free movement rights be affected?"

On the 24th of June this paper was admitted: "Brexit: what happens next?"

Now that the UK has voted to leave the EU, what will happen next? This Commons Library briefing paper looks at the immediate consequences of the vote and some of the longer term implications. This paper considers various questions about UK withdrawal from the EU and what is likely to happen in the coming weeks and months. The issues include the method of leaving the EU, continuing parliamentary scrutiny of EU business and the withdrawal negotiations, and the implications of Brexit for Scotland and Gibraltar.

Original Answer

Changes that will directly affect British nationals...

The British passport has "European Union" on the front cover. It will remain valid during the negotiations and a new design will most likely be phased in as existing passports expire;

The European Health Insurance Card (Ehic) (which enables British nationals to get access to medically necessary, state-provided healthcare during a temporary stay in any of the 28 EU countries, Iceland, Lichtenstein, Norway and Switzerland, under the same conditions and at the same cost (free in some countries) as people insured in that country), will remain valid during the negotiations and most likely be phased out as the UK negotiates separate treaties like those already in place with Australia and New Zealand;

The British driving permit will remain valid during the negotiations. Because it has an EU symbol on it, a new design will most likely be phased in as existing permits expire;

Spouses and long-term workers will most likely be grandfathered. There are no mass deportations of EU nationals envisioned.

See also: How will Brexit affect your finances?

Changes that will (or might affect) the family formation immigration routes...

A large part of the Brexit debate was focused on immigration. I will add a few things that are in the crosshairs. Indeed some of the rulings that help prompt the Brexit vote...

The Surinder Singh route. They never liked this ruling and it has acted as a thorn in the government's side for a long, long time. It's in the crosshairs and I think we can be sure that this immigration route will be closed, perhaps not this year, or even this decade, but it will be high on the list of priorities.

The Zambrano case. This is another case that never sat too well with the government. The government (two governments actually) were recalcitrant following the court's decision and it took a long time for them to even publish the guidance.

The Metock case. The UK's reaction was heated and recalcitrant but they finally implemented the ruling. Brexit advocates have seen this case as an extension of Singh (above) and hence a humiliating loss of sovereignty.

There are other rulings from the European Court that are similar and the UK has fought against them and opted out where they could.

These things affect boyfriends and girlfriends in long-distance relationships where they need an inward migration route because they cannot meet the rules. And overall these are about family formation and do not affect travellers and holiday makers to the UK.

Changes that affect tourists and visitors...

Remember that part of the mission of UK Visas and Immigration is to get lots of visitors in to the UK because the UK economy relies upon visitors, and this part of their mission will not change. If anything they will expand programmes like they are doing for China and India.

The UK has voted to leave the EU. How does this affect people traveling to the UK and vice-versa?

The outlook for tourists visiting the UK looks great! Sterling is at an all-time low so holidays will be cheaper, and visitors will be especially welcome because it boosts the economy.

Are other visa-free nationals (such as US/Canada citizens) affected?

No change. The UK has always operated at arm's length from the Schengen system. In about 4 or 5 years you will see the "EU Nationals" queue at the airport become rebranded into something that does the same thing but without the EU logo. And the EU rules about what you can carry and the HMRC customs declaration exits will be rebranded. It's reasonable to expect the government to adopt the EU customs rules into UK law. But this will (most likely) be done by statutory instrument so nobody will notice the change.

What other consequences might 'Brexit' have on travelers?

The first change a traveller is likely to see will be in the duty-free shops. Unless renegotiated, customs limits are likely to be introduced and of course any EU branding will be removed.

Update 25 June 2016

Laura Devine (a boutique immigration firm catering to high-net-worth individuals and frequent adviser to Parliament) posted this update yesterday...

Notes and comments...

Note: maybe you like the Singh Route, maybe you don't. Maybe you agree with what the government will do, maybe you don't. The point being that it's in the cross-hairs and there are strong public expectations that something will be done.

Side note: It's also reasonable to expect that EU citizens who have been here for a long time will be grandfathered in one way or another. Removals/deportations of people who are working in the economy is not in scope at all.

Comments: Addressing a comment from Relaxed (to whom thanks) on the feasibility of reversing the Singh ruling...

@JonathanReez I don't think so, and that's actually one of the few things we do know IMO. But that's a discussion we should have on the chat, the point is that assuming this kind of piecemeal adjustments are possible at all is highly speculative at this point. – Relaxed 11 hours ago

The practicalities and mechanics of actually reversing Singh are out-of-scope. The point being that the "immigration debate" surrounding Brexit was not about tourists and visitors.