Are there still parts of Burma that are off-limits to tourists?

Kayah State is now more opening especially for visiting Loikaw.
Rakhine is still restricted area near the border site.

There are some. Kayak State, Kachine state and most of the far northern area which has international borders.

However, it is possible to visit if you have the support of a local agency to get the visiting permit sometimes.

There are many places that are completely off travel in Myanmar (Burma). There are some places you can reach only via specific means of transportation (e.g. by air, or by boat). Then there are some that you can go to if you obtained the adequate permit from the MTT (Myanmar Travels and Tours, government agency) or if you are part of an organized tour with a travel agency.

The website of the MTT lists the specific places that are accessible within the restricted states/divisions : http://myanmartravelsandtours.com/tourist-information.htm

I don’t know if one can assume it to be completely up-to-date. In any case, the situation can change faster than the local immigration officers knowledge, so print this sheet out and show it to zealous local immigration officers that would like to turn you off.

This does not take into account recommendations from foreign governments (see Mark Mayo’s answer). Government advice against going to certain places may lead to your insurance company to refuse reimbursement should you land in trouble there.

According to Wikitravel:

Various insurgent groups continue to operate in the Shan, Mon, Chin
(Zomi), and Karen States of Myanmar, along the Thai and Chinese
borders. Travel to these regions generally requires a government
permit. The government also restricts travel to Kayah State, Rakhine
State and Kachin state due to insurgent activity. However travel is
entirely unrestricted to the districts of Yangon, Bago, Ayeyarwady,
Sagaing, Taninthayi, Mandalay and Magwe.

So it appears that according to that, at least, there are still restrictions.

However, there’s no date given, or source.

So next, we’ll go with the NZ Govt’s Travel Advisory page on Myanmar:

There is high risk to your security along the borders with China, Laos
and Thailand and we advise against all tourist and other non-essential
travel. Military activity, ethnic militias, armed drug smugglers, and
the presence of landmines pose a particularly high risk to your
safety. Travel restrictions are in place in most border areas and
there are only a limited number of legal crossing points.

There is high risk to your security in the state of Rakhine where the
government has imposed emergency measures following serious civil
unrest and we advise against all tourist and other non-essential
travel.

There is some risk to your security elsewhere in Myanmar due to the
unsettled political situation and threat from terrorism and we advise
caution.

This was current and updated today, so it seems that there are still some restrictions, although it’s sort of vague.

Back to Wikitravel, they actually list the restricted areas:

Much of Myanmar is closed to foreign travellers, and many land routes
to far-flung areas are also closed (for example, to Mrauk U, Kalewa,
Putao, Kengtung). Thus, while travellers can travel freely in the
Bamar majority Burmese heartland, travel tends to be restricted or
circumscribed in other places. In theory, any tourist can apply for a
permit to visit any restricted area or to travel on any restricted
land route. In practice, it is unlikely that any such permit will be
issued in a reasonable amount of time, or at all. Permit requests can
be made locally in some cases (for example, requests for the land
route to Kalewa can be made in Shwebo) but, in most cases, the request
has to be made in Yangon. Requests to visit restricted areas must be
made at the MTT (Myanmar Travel and Tours) office in Yangon (Number
77-91, Sule Pagoda Road, Yangon). Applications for local permits
can often be made at a local MTT office or at a police station. As of
writing this, local permits are available only for the following
places & routes:

Shwebo - Kalewa. A permit is necessary if going by road. It is uncertain whether one is required if going by boat.
Kengtung - Tachilek. This used to be straightforward but the availability is now uncertain.
Myitkyina - Indawgyi Lake. Easily available in Myitkyina but must travel with a guide. Your hotel or a local tour company can arrange

this for you.
Mrauk U Chin/ Zomi village tours. Easily available in Mrauk U but must visit with a guide. Your hotel or a local tour company can
arrange this for you.

All other permits must be obtained in Yangon.

Finally, the Foreign and Commonwealth Office from the UK has a page on the topic (they call it Burma), with some really good detail on the restrictions:

On 31 August the Burmese government announced restrictions on access
for tourists to Sittwe and Mrauk-U and a curfew remains in force
across much of Rakhine State. We advise against all but essential
travel to Rakhine State. The current curfew times are 18:00 to 06:00,
but this may change. We advise any British nationals in Rakhine State
to check timings of the curfew locally and follow any instructions.
The British Embassy in Rangoon stands ready to provide consular
assistance. British nationals working for NGOs and other companies
should keep in close contact with those organisations.

Following this outbreak of violence, a number of demonstrations took
place in Rangoon. While protests have not been violent, we advise that
you avoid all demonstrations and large gatherings.

The Burmese government restricts travel to most border areas. There
are a limited number of legal crossing points, but these could close
without notice:

Tachilek (Burma Shan State) – Mae Sai (northern Thailand border)
KawThoung (Burma Tanintharyi) – Ranong-Kawthoung (southern Thailand border)
Muse (Burma Shan State) – Ruili (China border)
Tamu (Burma Chin State) – Morei (India border)

You must exit Burma at the same border crossing from which you
entered, and Burmese immigration officers may request to hold your
passport until your visit is complete. Do not attempt to cross any
border illegally or enter restricted areas without the appropriate
permissions from the Burmese authorities. The Ministry of Hotels,
Tourism and Sport maintains a list of approved destinations. Tourists
can visit Rangoon, Mandalay, Bago and Irrawaddy regions without
restrictions. Other destinations are subject to limitations: access by
air or train but not by road. Queries on permission should be directed
to the Burmese authorities, not the British Embassy in Rangoon. See
www.myanmartourism.org or call the Ministry of Tourism on +95 67
406129.

Be particularly vigilant if travelling to border areas. There is
ongoing military activity close to borders with Thailand, Laos and
China especially in Shan, Karen and Kachin States. There is ongoing
conflict in parts of Kachin State and the far north of Shan State. The
Burmese authorities are currently restricting travel between Myitkyina
and Bhamo in Kachin state: travel to and between these two towns is
permitted by air only. There has been no fighting within the town of
Myitkyina itself, but town is experiencing occasional curfews as a
result of the conflict. Take local advice before travelling outside
Myitkyina within Kachin State, as the conflict situation remains
fluid. There has also been low-level fighting in central Shan State
and Karen State near the Thailand/Burma border in recent months.

Land mines also pose a threat in conflict areas.

You may only go to officially designated tourist areas. You need prior
permission from the tourism authorities for treks to remote parts of
the country. However, tourists have experienced difficulties with the
authorities even after obtaining such permission.