We're currently planning a trip to Japan. As we realize just now, it will cover the Obon week completely. To avoid getting stuck in the multitudes we try to shift our activities cleverly. What is the impact of Obon week on Japan?
First off, Obon is celebrated at different times of year in different parts of Japan. The 'main' date is around August 15, but Tokyo and parts nearby is July 15, and Shikoku, Okinawa and Chugoku still use the lunar calendar so their date slides around.
Second, Obon's kind of like Thanksgiving in the US, most people leave the cities and go visit their families in the countryside. The big travel peaks are thus away from the cities on the previous weekend (this year August 10-11) and back to them the next weekend (August 17-18). If you're going with the flow, it's very hard to find seats on planes, airplanes etc and the roads are jammed, but in the opposite direction it's quite feasible. Likewise, while hotels in popular tourist destinations are packed during Obon week, especially in cooler climes like Hokkaido, business hotels in boring bits of the city will be empty.
One saving grace is that, unlike New Year's, most businesses stay open for Obon so you won't have any problems visiting attractions, getting a bite to eat etc.
All that said, I would avoid going to Japan in August for another reason: it's intolerably hot and humid. Add in not only Obon but the school holidays that mean lots of locals are traveling as well for the entire month, and it's literally the worst possible time of the year to visit. I would retime your trip if at all possible.
There are multiple aspects to observe when dealing with holidays in Japan:
What does that mean for you?
The flights in and out of Japan as well as national travel are pretty much booked out. While one might think that there are open seats to travel to Japan when everyone is leaving the country, one has to consider that there are a lot of Japanese expats that will come home during this time and fill those incoming seats. Same goes of course for flight leaving Japan at the end of the holiday. As an example, many Japanese run restaurants in Hong Kong close over these holidays.
Since the dates are not as clear (as jpatokal lined out), it does not help to come/go a day earlier or later. You need to book your flights VERY early and in some cases also count on higher prices during that time.
Traveling to ANY countryside location becomes a nightmare. While there are still possibilities to get to day-trip locations very early and therefore evade the major rush, be prepared to meet literally millions of people on the road, in the trains and then on the destination itself. For example, going to Nikko by car during Obon will result in a 4+ hours traffic jam at the destination. Finding a parking place will be a nightmare by itself.
Also, as jpatokal mentioned, the cities are deserted. The big holidays (Golden week, obon, New year) are the only time in the year where you can enjoy empty streets and a releave from the omnipresent masses in the city. To give you a framework how strange this is for Japan, there is a photobook called "Tokyo Nobody" with photos of main streets in Tokyo - without any cars, taken on new year morning. If you show this book to Tokyo residents, you can see how amazed they are by looking at the photos since it is so rare to have that.
So going to the city during the holidays is something really recommendable - if you can get a flight. And jpatokal was right also that most shops are open. While some specialized, owner-run shops might close, most of the places are up and running. There is a huge opportunity to find exhibitions and locations that are normally booked out or have huge waiting times to visit during this time. But most of all, Tokyo itself becomes then finally a place where you can have a relaxed shopping experience.
Regarding the heat, Tokyo has a huge issue. Temperatures have been rising significantly in the city over the past 10 years since there is a significant higher amount of sky scrapers in Tokyo that block the airflow, specially the Shiodome complex had a major effect. Due to more sky scrapers which all of course operate air conditioners, which in return increase the temperature in the city. Temperatures above 30 degrees are completely normal.
One of the most filled places right now is for example the Tokyo Sky tree. I do not know the actual booking situation for this year Obon, but you might be lucky to get up there during this time. Also museums are a good place to go since you will have less waiting times, an air conditioned environment and you will have more time to look at the exposition.
So just as jpatokal mentioned, the conclusion is that a trip to Japan during this time might be a generally challenging visit. You will be able to enjoy Tokyo in a way that is normally not as enjoyable or even possible. If this will be your first and for some time only trip to Japan, I would recommend you to reschedule it.