Grading: The marfu' part of this hadith is "sahih li-ghayrihi" (authentic due to supporting reports), while the present chain is da'if.
Technical note / Defect ('illah): The reason is the unknown status (jahalah) of Muhammad bin al-Qasim, as he is not known. Al-Bukhari mentioned him in "al-Tarikh" (1/ 214) and Ibn Abi Hatim in "al-Jarh wa al-Ta'dil" (8/ 65), but they did not state his condition. Ibn Hibban mentioned him in "al-Thiqat", in keeping with his habit regarding unknown narrators. Salm bin Ibrahim is "da'if" and "muttaham" (accused), and al-Dhahabi declared him feeble in "al-Talkhis"; but he is corroborated.

Reference / Source: Thus al-Bukhari narrated it in "al-Tarikh al-Kabir" (1/ 214), Ya'qub bin Sufyan in his "Mashyakhah" (98), Abdullah bin Ahmad in his additions to his father's "al-Zuhd" (1019), Abu Ya'la in "al-Musnad al-Kabir" (as in "al-Matalib al-'Aliyah" 3226), al-Dulabi in "al-Kuna wa al-Asma'" (1538), al-Husayn bin Isma'il al-Mahamili in his "Amali" (narration of Ibn Mahdi al-Farisi 324), al-Tabarani in "al-Mu'jam al-Kabir" (14946), al-Bayhaqi in "Shu'ab al-Iman" (7850), al-Shajari in his "Amali" (2/ 219), Abu al-Qasim al-Asbahani in "al-Targhib wa al-Tarhib" (627 and 2358), Ibn Asakir in "Tarikh Dimashq" (2/ 132-133) and Diya' al-Din al-Maqdisi in "al-Mukhtarah" (9/ 424-426), via various routes from Ikrimah bin Ammar with this same chain.

Reference / Source: Abdullah bin al-Mubarak narrated in "al-Zuhd" (833), and via him Ibn Asakir (29/ 133-134), from Abdullah bin Lahi'ah, from Yazid bin Abi Habib, from Bukayr bin al-Ashajj, that Abdullah bin Salam came out of his garden carrying a bundle of firewood. When the people saw him they said: O Abu Yusuf! Among your children and slaves there are those who could do this for you! He said: I wanted to test my heart, whether it would find this objectionable (i.e., whether it was proud).
Grading: There is no harm in its men; it is Ibn al-Mubarak's narration from Ibn Lahi'ah, which the scholars have accepted.
Technical note / Defect ('illah): But Bukayr bin al-Ashajj (Bukayr bin Abdullah bin al-Ashajj) did not meet Abdullah bin Salam, so this narration of his is clearly "mursal".

Corroborating chains and supporting reports: Its marfu' part is supported by the hadith of Abdullah bin Mas'ud that passed under number (69), and it is in "Sahih Muslim" (91).