Are there any Mandarin dialects with three tones?
Lanyin Mandarin

特征
兰银官话古入声清音声母和次浊声母字今读去声，全浊声母今读阳平。其中永登、皋兰、古浪、天祝四个地点平声不分阴阳，只有平声、上声、去声三个调；其余地点阳平与上声同调，只有阴平、上声、去声三个调。

Ürümqi Mandarin (Included in the above Lanyin topolect)

乌鲁木齐话有3个单字调：
阴平 ˦˦ 44
阳平 ˥˩ 51
去声 ˨˩˧ 213

The following might not strictly be Mandarin, but are interesting nonetheless.
Tianshui Topolect (Gansu)

There are only 3 citation tones in Tianshui dialect.

Xunhua Topolect (Qinghai):

循化话有3个单字调，湟中话有2个单字调。

(The Huangzhong topolect only has two tones.)
Nanjiang Topolect

兰银官话北疆片、中原官话南疆片基本上都是3个单字调，但调类不同。