Which Macs cannot upgrade to Mountain Lion?
Apple said the following Macs will run Mountain Lion:


iMac (Mid 2007 or newer)
MacBook (Late 2008 Aluminum, or Early 2009 or newer)
MacBook Pro (Mid/Late 2007 or newer)
MacBook Air (Late 2008 or newer)
Mac mini (Early 2009 or newer)
Mac Pro (Early 2008 or newer)
Xserve (Early 2009)


So the complement of this list are not supported to run Mountain Lion (as far as I can tell; edit this list to add or subtract models as appropriate; you may be able to get it running on an unsupported machine):


Macintosh 128K
Macintosh 512K
Macintosh 128K (revised)
Macintosh XL
Macintosh Plus
Macintosh 512Ke
Macintosh Plus (Platinum)
Macintosh SE
Macintosh II
Macintosh IIx
Macintosh SE/30
Macintosh IIcx
Macintosh SE FDHD
Macintosh IIci
Macintosh Portable
Macintosh IIfx
Macintosh LC
Macintosh Classic
Macintosh IIsi
Macintosh Portable (backlit screen)
Macintosh Classic II
Quadra 700
Quadra 900
PowerBook 100
PowerBook 140
PowerBook 170
Macintosh LC II
Quadra 950
PowerBook 145
Macintosh IIvi
Macintosh IIvx
PowerBook 160
PowerBook 180
PowerBook Duo 210
PowerBook Duo 230
Macintosh LC III / III+
Macintosh Color Classic
Centris 610
Centris 650
Quadra 800
PowerBook 165c
Workgroup Server 80
Workgroup Server 95
PowerBook 145b
PowerBook 180c
Macintosh LC 520
Workgroup Server 60
Centris / Quadra 660AV
Quadra 840AV
PowerBook 165
Macintosh Color Classic II
Macintosh TV
Quadra 605
Quadra 610
Quadra 650
PowerBook Duo 250
PowerBook Duo 270c
Macintosh LC 550
Macintosh LC 575
Power Macintosh 6100
Power Macintosh 7100
Power Macintosh 8100
Workgroup Server 6150
Workgroup Server 8150
Workgroup Server 9150
PowerBook 520/c
PowerBook 540/c
PowerBook 550
PowerBook Duo 280
PowerBook Duo 280c
Quadra 630
PowerBook 150
Power Macintosh 6200 / 6300
Macintosh LC 580
Performa 5200
Power Macintosh 9500
Power Macintosh 7200
Power Macintosh 7500
Power Macintosh 8500
PowerBook 190
PowerBook 5300
PowerBook Duo 2300c
Workgroup Server 7250
Workgroup Server 8550
Performa 5260 / 5300
Performa 5400
Power Macintosh 7600
Performa 6360
Performa 6400
Power Macintosh 4400
PowerBook 1400
Power Macintosh 5500
Power Macintosh 6500
Power Macintosh 7300
Power Macintosh 8600
Power Macintosh 9600
PowerBook 3400
Twentieth Anniversary Macintosh
Workgroup Server 7350
Workgroup Server 9650
PowerBook 2400c
Power Macintosh G3 desktop
Power Macintosh G3 minitower
PowerBook G3
Power Macintosh G3 AIO
Macintosh Server G3
PowerBook G3 series
iMac G3
Power Macintosh G3 (Blue &amp; White)
Macintosh Server G3 (Blue &amp; White)
PowerBook G3 ("Lombard")
iBook
Macintosh Server G4
iMac (slot loading)
Power Mac G4 Graphite
PowerBook ("Pismo")
Power Mac G4 Cube
iBook (FireWire)
PowerBook G4 Titanium
iBook (white)
Power Mac G4 Quicksilver
Server G4 Quicksilver
iMac G4 15"
iBook (14")
eMac
Xserve
iMac G4 17"
Power Mac G4 MDD
Macintosh Server G4 MDD
PowerBook G4 Aluminum (12")
PowerBook G4 Aluminum (17")
Xserve slot loading
Xserve Cluster Node
Power Mac G5
PowerBook G4 Aluminum (15")
iBook G4 (12" / 14")
iMac G4 20"
Xserve G5
Xserve Cluster Node G5    Xserve
Power Mac G5 FX
iMac G5 17"
iMac G5 20"   iMac
Mac mini
Power Mac G5 dual core
iMac Core Duo
MacBook Pro (15")
Mac mini Core Solo
Mac mini Core Duo
MacBook Pro (17")
MacBook   (Late 2008 plastic)
Xserve (Intel) (Late 2006)
MacBook (Late 2006)
Mac mini Core 2 Duo
MacBook (Late 2007)
Xserve Harpertown
MacBook Penryn
1st generation MacBook Air  (early 2008)