Shaykh Su'uud Shuraim (Imaam and Khateeb of the Holy Masjid in Makkah) writes in his book ash-Shaamil concerning this issue:

The Scholars have differed with regards to this and there are two opinions.

1.

Raising your hands during the friday sermon is not allowed except during istisqaa (when the Imaam supplicates for rain).

This is the opinion of the Shaafi'is and the correct narration from the Hanaabilah aswell as Imaam az-Zuhree, Ibn Khuzaymah, Bayhaqi, an-Nawawi, Ibn Taymiyyah, Ibn al-Qayyim, Ibn Hajar and Shawkaani.

Both 'AbdiRazzaaq and Ibn Abee Shaybah reported in their Musannaf on the authority of az-Zuhree:

The raising of the hands during the friday sermon (during the supplication) is an innovation, and the first person to introduce it was 'Abdul Maalik.

2.

The raising of the hands during the friday sermon (for supplication) is allowed and there is no problem with that. This is the opinion of al-Bukhaaree  and some of the Maalikiyyah, and one of the reports of the Hanaabilah.

Imaam al-Bukhaaree made a chapter in his famous Saheeh giving it the title "Raising of the hands during the friday sermon" and reported the hadith of Anas bin Maalik wherein which the Prophet raised his hands when he made the istisqaa.

Conclusion:

After looking at the evidences the correct opinion seems to be - and Allah knows best - with those who say it is not allowed except for when the Imaam makes istisqaa.

And knowledge is with Allah.