This post summarizes the finds in this thread. Because the continent affiliation of some islands is a topic of debate, I list here flights involving islands, but not everyone may consider them a true inter-continental flight. Also, some operations are seasonal. Operations including islands and seasonal operations deserve being mentioned, but for each continent pair I will list going all the way to year-round landmass to landmass flights.
I will update this post as more short flights come in.
Africa-Asia
HRG-SSH 66 miles, landmass to landmass, year-round by EgyptAir
Africa-Europe
MLN-LEI 113 miles, landmass to landmass, year-round by Air Nostrum/Iberia
Africa-North America
SID-BOS 3329 miles, includes islands, year-round by TACV
CMN-YUL 3534 miles, landmass to landmass, year-round by Royal Air Maroc
Africa-Oceania
JNB-PER 5173 miles, landmass to landmass, year-round by South African
Africa-South America
SID-FOR 1767 miles, includes islands, year-round by TACV
LAD-GIG 3856 miles, landmass to landmass, year-round by TAAG
Asia-Europe
ADB-ATH 177 miles, landmass to landmass, year-round by Pegasus Airlines
Asia-North America
PKC-ANC 1958 miles, landmass to landmass, seasonal service by Yakutia Airlines
NRT-YVR 4674 miles, includes islands, year-round by Air Canada and JAL
ICN-YVR 5108 miles, landmass to landmass, year-round by Air Canada
Asia-Oceania
DIL-DRW 451 miles, includes islands, year-round by AirNorth and Timor Air
CGK-PER 1875 miles, includes islands, year-round by Jetstar
SIN-DRW 2074 miles, includes islands, year-round by Jetstar and Silk Air
KUL-PER 2563 miles, landmass to landmass, year-round by Malaysia
Asia-South America
DXB-GIG 7384 miles, landmass to landmass, year-round by Emirates
Europe-North America
FSP-YYT 172 miles, includes islands and a geopolitical oddity, year-round by Air Saint-Pierre
RKV-KUS 457 miles, includes islands and if you consider Greenland NA, year-round by Air Iceland
GOH-YFB 513 miles, includes islands and if you consider Greenland Europe, seasonal service by Air Greenland and FirstAir
UAK-CPH 2073 miles, includes islands and if you consider Greenland NA, year-round by Air Greenland
LHR-YYT 2316 miles, includes islands, seasonal service by Air Canada
PDL-BOS 2396 miles, includes islands, year-round by SATA International
LHR-YHZ 2857 miles, includes islands, year-round by Air Canada
OPO-EWR 3340 miles, landmass to landmass, year-round by TAP
Europe-Oceania
no flights (?)
Europe-South America
LIS-FOR 3478 miles, landmass to landmass, year-round by TAP
North America-Oceania
SFO-SYD 7417 miles, landmass to landmass, year-round by United
North America-South America
DAV-PAC 201 miles, landmass to landmass, year-round by Air Panama
Oceania-South America
AKL-SCL 6011 miles, includes islands, year-round by LAN
SYD-SCL 7060 miles, landmass to landmass, year-round by Qantas
Hopefully this map represent all the landmass-to-landmass links: