As you can see, there are some logos (slogans) of the word Allah (and other Allah's names) on many things such as flags. For instance there is a slogan of it on the flag of Islamic Republic of Iran. I wonder if it is Halal if we touch it without Wudhu (ablution)?
Touching of the word Allah in different matters apparently has the ruling of it on the papers. Then, touching this word would be haram without Wudhu.
But this could be considered from another aspect. Actually according to the Fatwa of Ayatollah Khamenei as a Marja al taqlid of Shiite:
Touching this logo (slogan of Iran) without wudhu would be haram in case it is considered as Esma-al-Jalalah from the Uruf, otherwise no problem. However it is better ignore touching it without wudu according to the precaution.
Peace be upon you
In the start, touching the flag of Allah's names without ablution is not forbidden and also touching of Quran without ablution is not forbidden but the better ablution before touching the Quran.