Several Japanese companies could claim the title, although depending upon how you count it they may be bested by the previous comment's Stella Artois.
The Mitsubishi Group is a quite famous corporation with an easy logo history- the company began in 1870 (145 years ago) with the name Mitsubishi, which is Japanese for "Three Water Chestnuts". The logo (three diamonds placed such that their points meet in the centre forming a triangle shape) is a picture of that name- in Japanese heraldry a water chestnut is depicted as a lozenge or diamond shape. The logo appears to date back to the very beginning of the company, as it was made by taking water chestnuts (the emblem of the Iwasaki clan, the family of its founder) and arranging them in the same pattern as that of the Yamanouchi clan's emblem's leaves (the Yamanouchi were a group of dignitaries in the region it was founded). Mitsubishi would thus take the 'oldest unmodified logo' from Twinnings Tea.
There are undoubtedly smaller family-held corporations in Japan whose logo is simply the family heraldry. These would date back to the adoption of the family crest, which could be hundreds of years ago. Without positive evidence for a particular one I would not claim any one in particular to be of great antiquity.
However, the oldest could quite possibly be the world's oldest corporation (or more accurately the oldest proto-corporation that became a modern corporation), The Kongo Group ( 金剛組 Kongōgumi). The Kongo Group was* a family-held corporation that was founded in 576 CE as a construction company specialising in Buddhist temples (thus its name, "The Vajra Group") whose origins date back to some of the earliest introductions of Buddhism to Japan. Their logo is simply one of the characters ( 剛 ) with half of a box around it (top and right sides), with a circle around it- it imitates the appearance of a stamp of that character. As the corporation dates back to the late iron age, the exact period the logo was adopted is quite difficult to discover. I don't know of anyone who has found a manufacterer's mark stamped on an 8th century buddhist temple, for instance, and most of their oldest woodwork cannot have survived without replacement. The use of such emblems took off in Japan in the 9th century CE, so I would not expect it to be any older than a thousand years.
*Kongogumi went bankrupt during the modern financial crisis in 2010, but was acquired by another company out of bankruptcy. I do not know whether one would thus use "Is" or "Was" for the group chartered in the 6th century by Prince Shotoku. The logo is still in use today for that subsidiary company.
EDIT: A now-deleted post suggested Stella Artois (a beer company, logo adopted 14th century but modified in the 20th century) and Twinnings Tea (mid-19th century, logo unchanged). References to these companies are in reference to the contention that those were the oldest.
Lyle 's have been using the same logo for their golden syrup since 1883. It's a dead lion surrounded by bees, which makes sense if you know your bible and you can read the explanation here...
http://www.lylesgoldensyrup.com/our-story/
It's not quite as old as Mitsubishi's 3 diamonds, but AFAIK it has remained the same throughout whereas Mitsubishi's has been subtly modified over the years.
There's a Dutch brewer 'Grolsch'. As far as I know their main logo has been unchanged since 1614.
It could also be 'Brand' beer. But if I'm not mistaken they revised their logo in 1814