Many areas of Asia, especially Southeast Asia and South Asia, are humid tropical, thus have certain risks attached to it -- such as dengue fever and risk of malaria.
Safety in Asia can vary wildly, but it is a safe place in general. Nearly all tourist attractions on the continent are far from conflict, but there are a few regions in which conflict and/or general lawlessness exists.
In Middle East
In Afghanistan and Iraq
The most obvious examples are Afghanistan and Iraq, two countries wholly under foreign occupation and in a state of war against insurgent groups, Taliban, Al Qaeda and numerous other armed forces. These countries are considered no-go areas and should be completely avoided by travellers (if one absolutely must go, consult War zone safety and the authorities of your country before you go). Yemen could also be added to the list of no-go countries, due to a very high threat of terrorist attacks, kidnappings, tribal violence and general lawlessness.
Other countries in Asia generally can be visited, though some countries have regions and areas that should be avoided. The Middle East is generally known for its political tensions, and while true for some part, most of the region can be visited without any major risks.
In Israel and Palestine
The Gaza Strip is effectively a war zone between Palestinian factions and the Israeli army, where kidnappings of foreigners have occurred. Israel has coped with missile attacks as well as suicide bombings by Islamic militant groups, such as Hamas and Hezbollah. Lebanon and the West Bank (Palestinian National Authority) generally cope with an unstable political situation and internal conflict. and Some regions of the Caucasus are considered dangerous due to active insurgent groups, particularly the North Caucasus (Chechnya), Abkhazia and South Ossetia.
In Iran and Pakistan
Iran and Pakistan cope with a low-level conflict in the Balochistan region against Baloch insurgents. More dangerously, Pakistan is active in a full-scale war in the North-West Frontier Province and the Federally Administered Tribal Areas, both of which should not be visited. Kashmir, claimed by both Pakistan and India, is also a region with tens of thousands of casualties since 1989 due to political strife and insurgency. In 2009, the Naxalite-Maoist insurgency in India has resurfaced, particularly in Chattishgarh and other parts of Eastern India. North-Eastern India also copes with dozens of insurgent groups, some of which have armed factions.
In Southeast Asia
Southeast Asia is a major travel region and most of it is perfectly safe to visit. A notable exception is East Timor, which continues to face sporadic internal ethnic and political tension and related violence may occur. But even in some popular countries, there are some areas that should be avoided. The Autonomous Region in Muslim Mindanao in the south of the Philippines is an area of conflict with several Muslim and communist rebel groups fighting with the Philippine army. It was declared the world's most hazardous area for journalists in 2009 by the Committee to Protect Journalists. 18 reporters died in a massacre in Maguindanao that claimed the lives of almost 60 people. The rest of the country specifically Luzon and Visayas islands are generally safe while the rest of Mindanao most specifically Davao and Cagayan de Oro areas are safe to travelers.
Thailand, the most visited country in Southeast Asia, is perfectly safe, with the notable exception of Deep Southern Thailand where fighting between the Thai military and Islamic insurgent groups still continues to this day.
Indonesia is also a very diverse country, which has led to armed groups fighting for independence in Papua. Maluku is relatively safe now, but has seen periodic eruptions of violence occur in 1999-2003. Indonesia also copes with Islamic extremist groups throughout the country. Bali, the most popular tourist destination of the country, dealt with Islamic terrorist attacks in Kuta in 2002 and 2005. The island can still be visited, but avoiding large gatherings and crowded night clubs could be advised.
In East Asia
East Asia is probably the safest area in the continent, but political tensions also exist in this region. You might want to avoid the border between North Korea and South Korea as these two countries are technically still at war with each other. China, a vast and diverse country, is surprisingly safe for its size, yet the restive provinces of Tibet (and its accompanying counties and prefectures in neighboring provinces) and Xinjiang are often restricted to foreign travelers at any hint of trouble.