It was stated by Tobi during his fight with Konan that Izanagi can only be used by those with both the DNA of Senju and Uchiha. We also know this is how the rinnegan is awakened, so why didn’t this happen?
Tobi was not able to awaken the Rinnegan despite having both Uchiha and Senju DNA because he was missing a certain step in the evolution, and that is, the Eternal Mangekyo Sharingan.
(I slightly explained obito's case in one of my previous answers)
The evolution of the Sharingan involves firstly awakening the Mangekyo, then after a combination with another family members Mangekyo, it evolves to an Eternal mangekyo sharingan, after which adding Senju DNA to the mix awakens the Rinnegan.
Tobi's Sharingan never reached the EMS and so thats why even tho he had combined both Uchiha and Senju DNA, he still was not able to awaken the Rinnegan.
At first, it was believed that the mixing of Senju and Uchiha DNA resulted in the Rinnegan but it was later in the series it was clarified that it's not necessarily the DNA of both that is needed but the chakra of the Sage of the Six Paths that awakens the Rinnegan.
This was done by Madara combining the chakra of the Sage's Sons Indra and Ashura who had been reincarnated into himself and Hashirama respectively. Of course the only people who awakened the Rinnegan (aside from the Sage) were Madara and Sasuke (who got his Six Paths Chakra directly from the Sage) who were both Uchiha with the Eternal Mangekyou Sharingan so it isn't known if the Sharingan or any of it's stronger forms is required for the process.
In the filler episodes that show the Sages Past, he awakened the Rinnegan immediately after unlocking the Mangekyou stage [There were no other Uchiha-Eyes to steal back then] but as it's filler it probably cannot be considered canon.
Tobi may have had some of Hashirama's Cells implanted through his prosthetic white side which likely contained some amount of Ashura's Chakra but he lacked Indra's Half of the Sages Chakra.
Additionally, it took Madara decades for his EMS and the eye which he sacrificed for Izanagi to transform into the Rinnegan so Tobi would perhaps not have had the time even if he somehow possessed Indra's Chakra.
Technically Izanagi doesn't require both the DNA for activation as mentioned here but to maintain it for a long time both the DNA are necessary.
According to me Obito doesn't have any direct intake of Hashirama's DNA. All he is able to do is sort of wear the Black Zetsu which doesn't lead to mixing of DNA.